Geophys. J. Int. (2020) 221, 129–141 doi: 10.1093/gji/ggz577 Advance Access publication 2019 December 31 GJI Rock and Mineral Physics, Rheology Low-temperature rheology of calcite Michael K. Sly, 1 Arashdeep S. Thind, 2 Rohan Mishra, 2,3 Katharine M. Flores 2,3 and Philip Skemer 1 1 Department of Earth and Planetary Sciences, Washington University in Saint Louis, 1 Brookings Dr, St. Louis, MO 63130, USA. E-mail: [email protected]2 Institute of Materials Science and Engineering, Washington University in Saint Louis, 1 Brookings Dr, St. Louis, MO 63130, USA 3 Department of Mechanical Engineering and Materials Science, Washington University in Saint Louis, 1 Brookings Dr, St. Louis, MO 63130, USA Accepted 2019 December 20. Received 2019 December 19; in original form 2019 April 24 SUMMARY Low-temperature plastic rheology of calcite plays a significant role in the dynamics of Earth’s crust. However, it is technically challenging to study plastic rheology at low temperatures because of the high confining pressures required to inhibit fracturing. Micromechanical tests, such as nanoindentation and micropillar compression, can provide insight into plastic rheology under these conditions because, due to the small scale, plastic deformation can be achieved at low temperatures without the need for secondary confinement. In this study, nanoinden- tation and micropillar compression experiments were performed on oriented grains within a polycrystalline sample of Carrara marble at temperatures ranging from 23 to 175 ◦ C, using a nanoindenter. Indentation hardness is acquired directly from nanoindentation experiments. These data are then used to calculate yield stress as a function of temperature using numerical approaches that model the stress state under the indenter. Indentation data are complemented by uniaxial micropillar compression experiments. Cylindrical micropillars ∼1 and ∼3 μm in diameter were fabricated using a focused ion beam-based micromachining technique. Yield stress in micropillar experiments is determined directly from the applied load and micropillar dimensions. Mechanical data are fit to constitutive flow laws for low-temperature plasticity and compared to extrapolations of similar flow laws from high-temperature experiments. This study also considered the effects of crystallographic orientation on yield stress in calcite. Although there is a clear orientation dependence to plastic yielding, this effect is relatively small in comparison to the influence of temperature. Key words: Creep and deformation; Plasticity, diffusion, and creep; Rheology: crust and lithosphere. 1 INTRODUCTION Calcite is a common rock-forming mineral in the upper crust and plays a major role in crustal deformation and rheology. While cal- cite may deform by one of several mechanisms there is abundant microstructural evidence for viscoplastic deformation at shallow conditions, within calcite-rich faults and shear zones. Over tem- peratures ranging from 100 to 500 ◦ C natural calcite exhibits de- formation microstructures that include twins and intracrystalline dislocations (Vernon 1981; Kennedy & White 2001; De Bresser et al. 2002; Liu et al. 2002; Rybacki et al. 2011; Wells et al. 2014; Kim et al. 2018; Negrini et al. 2018). At temperatures less than 400 ◦ C(T/T m ≈ 0.3), deformation of calcite is dominated by twinning (Ferrill et al. 2004). Twinning on e{10 ¯ 18}40 ¯ 41is common in calcite at low temperatures because it has a small critical resolved shear stress and is only weakly temperature-dependent (Burkhard 2000). However, if an individual crystal is unfavorably oriented for twinning, other slip systems, including slip on r{10 ¯ 14}¯ 2021and f{ ¯ 1012}2 ¯ 201, can be activated at low temperatures (De Bresser & Spiers 1997). There is also evidence of other deformation processes at low temperature such as crystal plasticity or recrystallization- accommodated dislocation creep (Vernon 1981; Kennedy & White 2001; De Bresser et al. 2002; Liu et al. 2002; Rybacki et al. 2011) and grain boundary sliding (Wells et al. 2014; Negrini et al. 2018). Numerous experiments have been performed to characterize the rheology of calcite over a wide range of pressure, temperature and stress states. Deformation experiments in compression and tension have been conducted on single crystals (Turner et al. 1954; Griggs et al. 1960; De Bresser & Spiers 1990, 1997; De Bresser et al. 1993; De Bresser 1996) and polycrystalline samples (Rowe & Rutter 1990; Rutter 1995; De Bresser 1996; Renner et al. 2002; Platt & De Bresser 2017), at temperatures of 20–1000 ◦ C and confining pressures of 0.2–1.0 GPa. Additional experiments in direct shear (Schmid et al. 1987; Verberne et al. 2013) and torsional geometries (Pieri, Burlini, et al. 2001; Pieri, Kunze, et al. 200; Barnhoorn et al. C The Author(s) 2019. Published by Oxford University Press on behalf of The Royal Astronomical Society. 129 Downloaded from https://academic.oup.com/gji/article-abstract/221/1/129/5691318 by Washington University, Law School Library user on 21 January 2020
Transcript
Geophys J Int (2020) 221 129ndash141 doi 101093gjiggz577Advance Access publication 2019 December 31GJI Rock and Mineral Physics Rheology
Low-temperature rheology of calcite
Michael K Sly1 Arashdeep S Thind2 Rohan Mishra23 Katharine M Flores23 andPhilip Skemer1
1Department of Earth and Planetary Sciences Washington University in Saint Louis 1 Brookings Dr St Louis MO 63130 USA E-mail mslywustledu2Institute of Materials Science and Engineering Washington University in Saint Louis 1 Brookings Dr St Louis MO 63130 USA3Department of Mechanical Engineering and Materials Science Washington University in Saint Louis 1 Brookings Dr St Louis MO 63130 USA
Accepted 2019 December 20 Received 2019 December 19 in original form 2019 April 24
S U M M A R YLow-temperature plastic rheology of calcite plays a significant role in the dynamics of Earthrsquoscrust However it is technically challenging to study plastic rheology at low temperaturesbecause of the high confining pressures required to inhibit fracturing Micromechanical testssuch as nanoindentation and micropillar compression can provide insight into plastic rheologyunder these conditions because due to the small scale plastic deformation can be achievedat low temperatures without the need for secondary confinement In this study nanoinden-tation and micropillar compression experiments were performed on oriented grains within apolycrystalline sample of Carrara marble at temperatures ranging from 23 to 175 C usinga nanoindenter Indentation hardness is acquired directly from nanoindentation experimentsThese data are then used to calculate yield stress as a function of temperature using numericalapproaches that model the stress state under the indenter Indentation data are complementedby uniaxial micropillar compression experiments Cylindrical micropillars sim1 and sim3 μm indiameter were fabricated using a focused ion beam-based micromachining technique Yieldstress in micropillar experiments is determined directly from the applied load and micropillardimensions Mechanical data are fit to constitutive flow laws for low-temperature plasticityand compared to extrapolations of similar flow laws from high-temperature experiments Thisstudy also considered the effects of crystallographic orientation on yield stress in calciteAlthough there is a clear orientation dependence to plastic yielding this effect is relativelysmall in comparison to the influence of temperature
Key words Creep and deformation Plasticity diffusion and creep Rheology crust andlithosphere
1 I N T RO D U C T I O N
Calcite is a common rock-forming mineral in the upper crust andplays a major role in crustal deformation and rheology While cal-cite may deform by one of several mechanisms there is abundantmicrostructural evidence for viscoplastic deformation at shallowconditions within calcite-rich faults and shear zones Over tem-peratures ranging from 100 to 500 C natural calcite exhibits de-formation microstructures that include twins and intracrystallinedislocations (Vernon 1981 Kennedy amp White 2001 De Bresseret al 2002 Liu et al 2002 Rybacki et al 2011 Wells et al 2014Kim et al 2018 Negrini et al 2018) At temperatures less than 400C (TTm asymp 03) deformation of calcite is dominated by twinning(Ferrill et al 2004) Twinning on e1018〈4041〉 is common incalcite at low temperatures because it has a small critical resolvedshear stress and is only weakly temperature-dependent (Burkhard2000) However if an individual crystal is unfavorably oriented fortwinning other slip systems including slip on r1014〈2021〉and
f1012〈2201〉 can be activated at low temperatures (De Bresser ampSpiers 1997) There is also evidence of other deformation processesat low temperature such as crystal plasticity or recrystallization-accommodated dislocation creep (Vernon 1981 Kennedy amp White2001 De Bresser et al 2002 Liu et al 2002 Rybacki et al2011) and grain boundary sliding (Wells et al 2014 Negriniet al 2018)
Numerous experiments have been performed to characterize therheology of calcite over a wide range of pressure temperature andstress states Deformation experiments in compression and tensionhave been conducted on single crystals (Turner et al 1954 Griggset al 1960 De Bresser amp Spiers 1990 1997 De Bresser et al1993 De Bresser 1996) and polycrystalline samples (Rowe amp Rutter1990 Rutter 1995 De Bresser 1996 Renner et al 2002 Platt ampDe Bresser 2017) at temperatures of 20ndash1000 C and confiningpressures of 02ndash10 GPa Additional experiments in direct shear(Schmid et al 1987 Verberne et al 2013) and torsional geometries(Pieri Burlini et al 2001 Pieri Kunze et al 200 Barnhoorn et al
Ccopy The Author(s) 2019 Published by Oxford University Press on behalf of The Royal Astronomical Society 129
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130 MK Sly et al
2004 Schuster et al 2017) have been conducted on polycrystallinesamples at temperatures of 25 to sim1500 C and confining pressuresof 02ndash60 GPa
Despite the microstructural evidence for viscoplastic deforma-tion in natural calcite samples deformed at shallow crustal depths(corresponding to temperatures of 100ndash500 C) most laboratoryexperiments have been performed at temperatures greater than 300C Indeed only a handful of studies report results from experi-ments below 300 C (Turner et al 1954 Schmid et al 1987 Roweamp Rutter 1990 Verberne et al 2013 Schuster et al 2017) Hightemperatures are commonly used in rock deformation experimentsto promote thermally activated processes including diffusion anddislocation creep at laboratory strain rates that are greater than tec-tonic strain rates High pressures are also needed to suppress frac-turing Low-temperature flow laws are therefore mainly constrainedthrough the extrapolation of data from higher temperatures (eg DeBresser amp Spiers 1997 De Bresser et al 2002 Renner et al 2002Mei et al 2010) This extrapolation increases the uncertainty inany modeling or inference of deformation conditions of calcite-richrocks deformed in shallow crustal faults and shear zones (eg Hirthamp Kohlstedt 2015)
In this study we determine directly the rheology of calcite at lowtemperatures (23ndash175 C) using micromechanical methods includ-ing instrumented nanoindentation and micropillar compression Ina nanoindentation experiment a sharp probe is pushed into thematerial of interest at a constant loading rate or a constant displace-ment rate referred to as load-control and displacement-controlrespectively (Oliver amp Pharr 1992 2004 VanLandingham 2003)Simultaneous records of load and probe displacement are used todetermine the mechanical response of the sample Nanoindentationfacilitates the study of plastic deformation in geologic materialsat low temperatures because the self-confined nature of the exper-iment suppresses fracturing (Goldsby et al 2004 Kearney et al2006 Kranjc et al 2016 Kumamoto et al 2017 Thom amp Goldsby2019) Nanoindentation experiments are also quick in comparisonto other rock deformation experiments facilitating the collection oflarge data sets (eg gt2000 individual deformation experiments inthis study) However the complex stress state below the indentertip presents a challenge when uniaxial properties are desired ne-cessitating a numerical analysis to obtain material properties suchas yield stress
Nanoindentation can be complemented by additional micropillarcompression tests which enable the direct measurement of uniaxialyield stress at length scales similar to those in nanoindentationexperiments In a micropillar compression experiment a micron-scale pillar is micromachined using a focused ion beam (FIB) andthen deformed in uniaxial compression using a flat probe (Uchicet al 2009) Due to the small size of the pillar brittle processes aresuppressed and viscoplastic deformation may be observed (Korteamp Clegg 2009 Korte-Kerzel 2017) In this study we use data fromboth nanoindentation and micropillar compression experiments toelucidate the rheology of calcite at low temperatures that are relevantto geological studies and geodynamic modeling of crustal rheologyand orogenesis
2 M E T H O D S
21 Specimen preparation and initial characterization
Indentation experiments were conducted within individual grainsof a single polycrystalline sample of Carrara marble which has
Figure 1 Equal angle upper hemisphere antipodal pole figure showing thecrystallographic orientations used in this study Colors correspond to crystalorientation and are the same colors used in Figs 3ndash6 All grains chosen fortesting are oriented such that one of the directions shown here is within 10of the indentation direction
an average grain size of 100ndash200 μm A 7 times 9 times 1 mm sam-ple was cut from a larger block that has been widely used in rockdeformation studies (eg Fredrich et al 1989 1990 Xu amp Evans2010) and polished using progressively finer diamond grit witha final polish using 002 μm colloidal silica Crystallographic ori-entations of individual grains were determined using an OxfordInstruments Electron Backscatter Diffraction (EBSD) system ona JEOL 7001 FLV scanning electron microscope (SEM) operat-ing at an accelerating voltage of 20 kV and working distance of19 mm For this study grain orientation is defined by the crys-tallographic direction parallel to the direction of indentation Theseorientations were selected to maintain consistency with previous ex-periments by Turner et al (1954) and De Bresser amp Spiers (1997)both of which investigated the orientation dependence of twinningin calcite Five crystallographic orientations were chosen for in-dentation experiments [0001] 〈1120〉 〈1010〉 〈2243〉 and 〈4041〉(Fig 1) [0001] corresponds to the c-axis of the calcite grain while〈1120〉 and 〈1010〉 correspond to the a-axes and the poles to mir-ror planes respectively these two orientations are normal to thec-axis Crystals deformed parallel or subparallel to [0001] are ori-ented unfavorably for twinning while crystals deformed normalto the c-axis (〈1120〉 and 〈1010〉) are oriented favorably for twin-ning on e1018〈4041〉 under uniaxial compression (Turner et al1954 De Bresser amp Spiers 1997) The two remaining orientations〈2243〉 and 〈4041〉 are also unfavorable for twinning under uniax-ial compression (De Bresser amp Spiers 1997) Twenty-four grainsincluding four to five grains in each orientation defined above werechosen for testing Due to the weak texture of Carrara marble andthe paucity of ideal matches to the desired orientations the grainschosen for this study are within 10 of a desired orientation We donot account for the azimuthal orientation of the grain with respectto the faces of the three-sided pyramidal indenter Differences inazimuthal orientation of grains may introduce small scatter in ourresults which is on the order of 5
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Low-temperature rheology of calcite 131
Figure 2 Load-displacement curves for high quality (blue) and poor quality(red) indentations from the same grain at the same temperature Typical loaddisplacement curves have continuous curvature and occasionally show fewsmall increases in displacement (sim1ndash10 nm) referred to as pop ins A smallchange in displacement at the maximum load results from creep during the2 s hold in the loading function Load functions that deviate from this shapeoften imply that the data are not useful and are not used in this study
22 Nanoindentation
Load-controlled nanoindentation experiments in this study wereperformed using a diamond Berkovich indenter probe in a HysitronTI 950 TriboIndenter equipped with a temperature control stageIndividual indentation experiments consist of three segments 5 sof loading at a constant rate to the maximum load of 5 mN a 2-shold at the maximum load and 5 s of unloading at a constant rateWe use a maximum load of 5 mN to ensure contact depths aresufficiently deep to avoid complications with misshapen or bluntedindenter tips The indentation load P and probe displacement hare recorded simultaneously during each indentation experiment(Fig 2) Load measurements have a nominal precision of lt1 nNand displacement measurements have a precision of lt002 nm
The probe displacement is then converted to contact depth hc foreach indent using the maximum load applied Pmax and the initialunloading contact stiffness S which is determined from the slopeof the initial portion of the unloading curve
hc = h minus 075Pmax
S(1)
Contact area is related to contact depth through a polynomialfunction of the form
Ac = a0h2c + a1hc + a2h12
c + a3h14c + (2)
where ai are constants determined by indenting a material withwell-known indentation hardness and reduced elastic modulus overa wide range of loads Fused silica was indented over the range of100ndash10 000 μN for this calibration Indentation hardness H andreduced elastic modulus Er are calculated for each indent usingeqs (3) and (4) respectively and compared to the known values todetermine the accuracy of the area function
H = Pmax
Ac(3)
Er = Sradic
π
2radic
Ac
(4)
The area function is considered acceptable if the calculated valuesare within 10 of the known values
The measured load and displacement along with the contact areaare used to calculate the indentation hardness and the reduced elasticmodulus of the indented material The reduced elastic modulus isrelated to Youngrsquos modulus by
1
Er=
(1 minus ν2
E
)sample
+(
1 minus ν2
E
)indenter
(5)
where E is Youngrsquos modulus and ν is Poissonrsquos ratio and the sub-scripts refer to the properties of the sample and the indenter tip TheYoungrsquos modulus and Poissonrsquos ratio for the diamond indenter are1140 GPa and 007 respectively
The shape of the loadndashdisplacement curve is used to assess thequality of the data (Fig 2) Indents with atypical loadndashdisplacementcurves may indicate that the probe tip encountered something otherthan the flat sample surface (eg defects debris on the sample orsurface roughness) and are excluded from subsequent analysis
Between 20 and 25 indents were made in each grain at fourtemperatures (23 75 125 and 175 C) resulting in approximately100 indents in each grain and over 2000 indents in total The vol-ume affected by each indent extends beyond the contacted area andindividual indents were spaced 10 μm apart to avoid interactionbetween indents For experiments at all temperatures the samplewas clamped to the stage and allowed to equilibrate to the desiredtemperature for at least 30 min prior to the start of the experimentAfter the indents were completed the structure and morphologyof individual indents were examined in situ using Scanning ProbeMicroscopy (SPM) in the Hysitron TI 950 TriboIndenter as well asex situ using SEM SPM produces topographical images by scan-ning the nanoindenter probe across the sample while maintaininga constant load between the probe and the surface SPM imageswere taken of representative indentations in each orientation Usingthe SEM secondary electron images were taken of representativeindents produced at each temperature and orientation
23 Micropillar compression
Micropillars were fabricated using a focused ion beam (FIB) mi-cromilling technique that was originally developed and applied tometallic materials (Uchic et al 2009) Two instruments were used aZeiss Crossbeam 540 FIB-SEM and an FEI Quanta 3D FEG Priorto fabricating micropillars the entire Carrara marble sample wascoated with iridium to avoid sample charging The Ga+ ion beamwas then used to sputter away small amounts of the sample materialin concentric annular rings to produce a well with a cylindrical pil-lar at the centre To minimize ion implantation damage the pillarsare machined using progressively smaller concentric rings with pro-gressively lower ion beam doses to produce a 25 μm diameter wellwith a 1 or 3 μm diameter pillar in the centre The micropillars havean approximate diameter to height ratio of 12 Multiple micropil-lars were made in several grains oriented for compression parallelto 〈2243〉 (unfavorably oriented for twinning) and for compressionparallel to 〈1010〉 and 〈1120〉 (favorably oriented for twinning)
Micropillar compression experiments were conducted in loadcontrol using monotonic stair-step loading functions with alternat-ing segments of loading and holding Experiments were stopped
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132 MK Sly et al
when a sharp change in displacement with respect to time was ob-served which was assumed to coincide with the onset of brittlefailure or plastic yielding The load at which yielding occurred isconverted to yield stress by
σy = Py
Ap (6)
where Py is the load at the time of failure or yielding and Ap is thearea of the top of the micropillar prior to testing
24 Scanning transmission electron microscopy
A thin foil from a deformed micropillar was prepared using a Hi-tachi NB5000 focused ion and electron beam system The top of themicropillar was coated with a 1-μm-thick layer of carbon followedby a 1-μm-thick layer of platinum Additionally a 2-μm-thick plat-inum coating was deposited to protect both sides of the micropillarfrom collapsing during FIB lift-out A 20 kV beam with 07 nAcurrent was used to cut the foil from the base of the micropillarwhile beam currents of 007 nA at 10 kV and 001 nA at 5 kVwere used to perform rough and fine milling respectively to makethe foil electron transparent The resulting foil was mounted ona Cu grid for scanning transmission electron microscopy (STEM)experiments
STEM imaging was carried out using the aberration-correctedNion UltraSTEMTM 200 (operating at 200 and 100 kV) at OakRidge National Laboratory which is equipped with a fifth-orderaberration corrector and a cold-field emission electron gun TheCu grid was baked at 160 C under vacuum prior to the STEMexperiments to remove organics
3 R E S U LT S
31 Nanoindentation
The indentation hardness of calcite varies systematically with crys-tallographic orientation (Table 1 and Fig 3) Grains indented parallel([0001]) and subparallel (〈2243〉) to the c-axis are harder than thoseindented normal to the c-axis (〈1120〉 and 〈1010〉) The fifth ori-entation tested (〈4041〉) which lies between the c-axis [0001] and〈1010〉 exhibits intermediate hardness Differences between grainsof similar orientations are assumed to be due to small misalignmentsof grains with respect to the indenter
At room temperature (23 C) grains with hard orientations par-allel or subparallel to the c-axis have an average hardness of282 plusmn 012ndash286 plusmn 013 GPa Grains in softer orientations have anaverage hardness of 259 plusmn 010ndash260 plusmn 009 GPa Anisotropy ofhardness AH is defined as
AH =(
Hmax minus Hmin
Hall
)times 100 (7)
where Hmax is the average hardness of the grains in the strongestorientation Hmin is the average hardness of the grains in the weak-est orientation and Hall is the average hardness computed for allindents at a given temperature Hmax is always the average hardnessof the grains indented parallel or subparallel to the c-axis ([0001]or 〈2243〉) Hmin is always the average hardness of the grains in-dented in one of the two orientations normal to the c-axis (〈1120〉 or〈1010〉) At room temperature the anisotropy of hardness is 100
Indentation hardness of calcite is also temperature dependent(Table 1 and Fig 3) For orientations [0001] and 〈2243〉 the hardness
decreases to 216 plusmn 008 and 217 plusmn 005 GPa respectively at 175C Similarly the hardness in the softer orientations 〈1120〉 and〈1010〉 decreases to 198 plusmn 011 and 198 plusmn 008 GPa respectivelyat 175 C The anisotropy of hardness at 175 C is similar to that atroom temperature with a value of 92
Fig 4 shows representative secondary electron images of the tri-angular indents produced by the three-sided pyramidal Berkovichprobe Indent size and depth both increase with temperature Theapothem and depth of room temperature indents are approximately500 nm and 240ndash280 nm respectively The indents increase insize to an apothem of 750 nm and depths of 280ndash330 nm at175 C At lower temperatures the sides of indents performedin grains oriented with the indentation axis parallel or subparal-lel to the c-axis are rough and cracks are present The most extremecracking appears in grains indented parallel to the c-axis at roomtemperature (Fig 4) As temperature increases the number anddensity of cracks decreases and the sides of the indents becomesmoother There are only a few cracks in indents performed at125 C and no cracks observed in indents performed at 175 CThere is positive relief referred to as lsquopile-uprsquo around the sidesof the indents formed by material that is displaced during inden-tation SPM images in Fig 5 show additional examples of thisfeature
32 Micropillar compression
Fourteen micropillar compression experiments performed on pil-lars with a nominal diameter of 1 μm and oriented unfavorablyfor twinning (compression parallel to 〈2243〉) exhibit yield stressesvarying from 082 to 115 GPa at 23 C and 055ndash077 GPa at 175C Pillars were examined before and after experiments to iden-tify the mode of failure (Table 2) All pillars were characterizedas having failed by brittle fracture diffuse ductile deformation orvia deformation along shear bands Brittle fracture was identifiedby the presence of open cracks Diffuse ductile deformation wasidentified by a change in the shape of the pillar without any associ-ated cracking or deformation along shear bands Shear bands wereidentified in secondary electron images as narrow bands of highcontrast
Twelve 1 μm diameter pillars tested in grains oriented favorablyfor twinning (compression parallel to 〈1010〉) were more variablewith yield stresses ranging from 056 to 103 GPa at 23 C and 027ndash072 GPa at 175 C All of the 1 μm pillars show either fractureor diffuse ductile deformation without any noticeable shear bands(Table 2)
Additional micropillar compression experiments were conductedon 12 pillars with a diameter of 3 μm in both unfavorable (com-pression parallel to 〈2243〉) and favourable (compression parallelto 〈1010〉 and 〈1120〉)orientations for twinning The yield stressesof the 6 pillars in the unfavorable orientation are comparable tothose of the 1 μm pillars in the same orientation ranging from073 to 115 GPa However the 6 pillars tested in favourable ori-entations for twinning are markedly weaker than the 1 μm pillarstested under similar conditions with yield stresses ranging from010 to 036 GPa The 3 μm pillars exhibit notably different de-formation features from the 1 μm diameter pillars While brittleand diffuse plastic deformation are observed in the 3 μm pillarsthe larger samples also exhibit shear bands in the pillars favorablyoriented for twinning that are not apparent in pillars unfavorably ori-ented for twinning (Fig 6) STEM observation of these shear bandsindicates that they are deformation twins on e1018 Fig 7(a)
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Low-temperature rheology of calcite 133
Table 1 Indentation hardness as a function of orientation and temperature
Figure 3 Indentation hardness plotted against temperature Small trianglesrepresent the individual indentations (n = 2183) and large triangles rep-resent averages for 20ndash25 indentations performed in a single grain (colourcorresponds to the orientation of the grain following the colour scheme inFig 1) Error bars in the bottom of the figure which are offset for clarityshow plusmn one standard deviation for each orientation
shows a medium-angle annular dark field (MAADF) image of pil-lar 2086 P02 which is oriented nominally with the m1010planeas its surface with a shear band traversing across the pillar from top-left to bottom-right at an angle of sim59 from the horizontal Thisangle is comparable to the known angle between m and e of sim64Fig 7(b) shows a high-angle annular dark field (HAADF) image ofthe pillar with shear band highlighted as Region II and either sidesof the shear band highlighted as Regions I and III Fig 7(c) showsthe fast Fourier transform (FFT) diffraction patterns of Regions IndashIII We have used the crystal structure of calcite from Chessin et al(1965) to index the FFT patterns The FFT patterns confirm twin-ning during deformation with an in-plane rotation of 4378 for theshear band (Region II) with respect to the undeformed crystal oneither side (Region I and II) This angle is close to the theoreticalangle of 525 (Barber amp Wenk 1979) The differences between theangles measured here and the theoretical angles are likely becausethe STEM foil was not cut perfectly orthogonal to the twin Theout of plane direction corresponds to [2201] orientation of calcitecrystal structure (Chessin et al 1965)
4 D I S C U S S I O N
41 Yield stress from nanoindentation experiments
While indentation hardness measurements are precise and relativelystraightforward to collect there remains a significant challenge toscale these values to uniaxial properties used in traditional consti-tutive flow laws and models of deformation To relate indentationhardness measurements to uniaxial mechanical properties indenta-tion hardness must be converted to yield stress σy This relationshipis commonly expressed by a simple linear function where hardnessis proportional to yield stress via the constraint factor C (eq 8)(Tabor 1970)
H = Cσy (8)
The constraint factor may vary from 11 to 30 where 11 repre-sents the elastic limit and 30 represents the plastic limit (Johnson1970 Evans amp Goetze 1979) Constraint factors are typically around3 for materials with high Eσy ratios (gt133) such as metals and canbe as low as 15 for materials with low Eσy ratios such as glassesand polymers (Swain amp Hagan 1976 Fischer-Cripps 2011 Shawamp DeSalvo 2012) Calcite has a relatively low Eσy ratio whichmeans that the constraint factor is predicted to be in the range ofC = 15ndash30
Multiple models have been used to estimate the constraint factorand yield stress Here we present calculations for four such modelsJohnson (1970) Evans amp Goetze (1979) Mata et al (2002) Mataamp Alcala (2003) and Ginder et al (2018) (Table 3) As we haveno independent constraint on the yield stress at these deformationconditions we make no determination that one model is more validthan the others
The first of these models derived by Johnson (1970) comparesindentation to an expanding cavity in an elasticndashplastic solid Thismethod states that the ratio of indentation hardness to yield stress isdetermined by the geometry of the indenter and the elastic modulusof the material
H
σy= 2
3
[1 + ln
( E tan(θ)σy
+ 4 (1 minus 2ν)
6 (1 minus ν)
)] (9)
Here H is indentation hardness σy is yield stress E is Youngrsquosmodulus ν is Poissonrsquos ratio and θ is the angle between the surfaceof a conical indenter and the indented surface The value for θ
can be adjusted to compensate for different indenter tip geometriesby relating the volume of material displaced by differently shapedindenter probes (Fischer-Cripps 2011) We use eq (9) to determineyield stress though it can be simplified to eq (10) if a Poissonrsquosratio of 05 (ie an incompressible material) is assumed
H
σy= 2
3
[1 + ln
(1
3
E tan (θ )
σy
)] (10)
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134 MK Sly et al
Figure 4 Array of representative secondary electron images of indentations at each temperature step (rows) and in each orientation (columns) Images arecolored based on orientation following the colour scheme in Fig 1 The size of the indentations increases with increasing temperature apothems increase from500 to 750 nm and depths increase from 240 to 330 nm from 23 to 175 C Linear cracks are present in each of the four images of indentations performedparallel to 〈4041〉 (marked by dashed arrows) All of these features appear in the same orientation but do not appear to align with twin planes The imagesof these indentations were taken after the sample was coated with iridium in preparation for micropillar fabrication and these features are interpreted to becracks or tears in the iridium coating caused by the SEM electron beam There are bright linear features present in the lower right-hand face of the indentationsperformed parallel to 〈1010〉 and dark linear features present in the upper face of the indentations performed parallel to 〈1120〉 though these are less prominent(marked by solid arrows) Due to the orientation of these grains and the fact that these features are not present in other grains it is thought that these featuresare twin planes
Figure 5 Scanning probe microscope (SPM) images of indentations madeat room temperature in each orientation tested Images are colored basedon orientation following the colour scheme in Fig 1 Pile up is observed in4 of the 5 orientations (solid arrows) in varying amounts and on differentsides of the indentation This variation is due to the anisotropy of the calciteFractures are observed on the surface of the indentation parallel to [0001](dashed arrow)
Evans amp Goetze (1979) determine an empirical model based onthe expanding cavity model of Johnson (1970) using microindenta-tion data in olivine This model shows the same relationship betweenyield stress and indentation hardness as Johnsonrsquos model though
with slightly different coefficients
H
σy= 019 + 16 log
(E tan (θ )
σy
) (11)
Eq (11) is used in combination with Taborrsquos eq (8) to solve foryield stress and constraint factor simultaneously
Mata et al (2002) and Mata amp Alcala (2003) use finite elementanalysis to develop a new model for indentation in metals Mataamp Alcala use Johnsonrsquos idea that the constraint factor is a func-tion of the term ln(Eσr ) where σr is the reference stress at 10strain They then use finite element analysis to determine a polyno-mial function of ln(Eσr ) which is used to solve for the referencestress
H
σr=
sum4
i=0ci
[ln
(E
σr
)]i
(12)
Here ci are constants determined by the finite element analysisThis reference stress σr is comparable to the yield stresses deter-mined by Johnson (1970) Evans amp Goetze (1979) and Ginderet al (2018) The reference stress is then used in conjunction withthe curvature of the indentation load-displacement curve to deter-mine a pile-up factor and work hardening exponent that are usedto calculate yield stress Due to these additional steps the yieldstress determined by Mata amp Alcala is approximately a factor of4 lower than the reference stress However because the referencestress from Mata amp Alcala (2003) is defined the same way as theyield stresses from the other models we use it in subsequent com-parisons This model was initially developed for materials with yieldstrengths between 50 and 1000 MPa and Youngrsquos moduli between70 and 200 GPa The results from our study show that yield stress for
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ashington University Law
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Low-temperature rheology of calcite 135
Table 2 Micropillar compression results
calcite as defined in Mata amp Alcala (2003) (sim250 Mpa) is withinthe validated range and that the Youngrsquos modulus (50ndash65 GPa de-pending on temperature and orientation) is just below the validatedrange
Finally Ginder et al (2018) developed a model similar to thatof Johnson (1970) but for a power-law creeping solid as opposedto an elasticndashplastic solid This method uses Johnsonrsquos analysis ofthe dependence of indentation hardness on yield stress and Youngrsquos
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ashington University Law
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136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
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ashington University Law
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Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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ashington University Law
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Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
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140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
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nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
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nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
130 MK Sly et al
2004 Schuster et al 2017) have been conducted on polycrystallinesamples at temperatures of 25 to sim1500 C and confining pressuresof 02ndash60 GPa
Despite the microstructural evidence for viscoplastic deforma-tion in natural calcite samples deformed at shallow crustal depths(corresponding to temperatures of 100ndash500 C) most laboratoryexperiments have been performed at temperatures greater than 300C Indeed only a handful of studies report results from experi-ments below 300 C (Turner et al 1954 Schmid et al 1987 Roweamp Rutter 1990 Verberne et al 2013 Schuster et al 2017) Hightemperatures are commonly used in rock deformation experimentsto promote thermally activated processes including diffusion anddislocation creep at laboratory strain rates that are greater than tec-tonic strain rates High pressures are also needed to suppress frac-turing Low-temperature flow laws are therefore mainly constrainedthrough the extrapolation of data from higher temperatures (eg DeBresser amp Spiers 1997 De Bresser et al 2002 Renner et al 2002Mei et al 2010) This extrapolation increases the uncertainty inany modeling or inference of deformation conditions of calcite-richrocks deformed in shallow crustal faults and shear zones (eg Hirthamp Kohlstedt 2015)
In this study we determine directly the rheology of calcite at lowtemperatures (23ndash175 C) using micromechanical methods includ-ing instrumented nanoindentation and micropillar compression Ina nanoindentation experiment a sharp probe is pushed into thematerial of interest at a constant loading rate or a constant displace-ment rate referred to as load-control and displacement-controlrespectively (Oliver amp Pharr 1992 2004 VanLandingham 2003)Simultaneous records of load and probe displacement are used todetermine the mechanical response of the sample Nanoindentationfacilitates the study of plastic deformation in geologic materialsat low temperatures because the self-confined nature of the exper-iment suppresses fracturing (Goldsby et al 2004 Kearney et al2006 Kranjc et al 2016 Kumamoto et al 2017 Thom amp Goldsby2019) Nanoindentation experiments are also quick in comparisonto other rock deformation experiments facilitating the collection oflarge data sets (eg gt2000 individual deformation experiments inthis study) However the complex stress state below the indentertip presents a challenge when uniaxial properties are desired ne-cessitating a numerical analysis to obtain material properties suchas yield stress
Nanoindentation can be complemented by additional micropillarcompression tests which enable the direct measurement of uniaxialyield stress at length scales similar to those in nanoindentationexperiments In a micropillar compression experiment a micron-scale pillar is micromachined using a focused ion beam (FIB) andthen deformed in uniaxial compression using a flat probe (Uchicet al 2009) Due to the small size of the pillar brittle processes aresuppressed and viscoplastic deformation may be observed (Korteamp Clegg 2009 Korte-Kerzel 2017) In this study we use data fromboth nanoindentation and micropillar compression experiments toelucidate the rheology of calcite at low temperatures that are relevantto geological studies and geodynamic modeling of crustal rheologyand orogenesis
2 M E T H O D S
21 Specimen preparation and initial characterization
Indentation experiments were conducted within individual grainsof a single polycrystalline sample of Carrara marble which has
Figure 1 Equal angle upper hemisphere antipodal pole figure showing thecrystallographic orientations used in this study Colors correspond to crystalorientation and are the same colors used in Figs 3ndash6 All grains chosen fortesting are oriented such that one of the directions shown here is within 10of the indentation direction
an average grain size of 100ndash200 μm A 7 times 9 times 1 mm sam-ple was cut from a larger block that has been widely used in rockdeformation studies (eg Fredrich et al 1989 1990 Xu amp Evans2010) and polished using progressively finer diamond grit witha final polish using 002 μm colloidal silica Crystallographic ori-entations of individual grains were determined using an OxfordInstruments Electron Backscatter Diffraction (EBSD) system ona JEOL 7001 FLV scanning electron microscope (SEM) operat-ing at an accelerating voltage of 20 kV and working distance of19 mm For this study grain orientation is defined by the crys-tallographic direction parallel to the direction of indentation Theseorientations were selected to maintain consistency with previous ex-periments by Turner et al (1954) and De Bresser amp Spiers (1997)both of which investigated the orientation dependence of twinningin calcite Five crystallographic orientations were chosen for in-dentation experiments [0001] 〈1120〉 〈1010〉 〈2243〉 and 〈4041〉(Fig 1) [0001] corresponds to the c-axis of the calcite grain while〈1120〉 and 〈1010〉 correspond to the a-axes and the poles to mir-ror planes respectively these two orientations are normal to thec-axis Crystals deformed parallel or subparallel to [0001] are ori-ented unfavorably for twinning while crystals deformed normalto the c-axis (〈1120〉 and 〈1010〉) are oriented favorably for twin-ning on e1018〈4041〉 under uniaxial compression (Turner et al1954 De Bresser amp Spiers 1997) The two remaining orientations〈2243〉 and 〈4041〉 are also unfavorable for twinning under uniax-ial compression (De Bresser amp Spiers 1997) Twenty-four grainsincluding four to five grains in each orientation defined above werechosen for testing Due to the weak texture of Carrara marble andthe paucity of ideal matches to the desired orientations the grainschosen for this study are within 10 of a desired orientation We donot account for the azimuthal orientation of the grain with respectto the faces of the three-sided pyramidal indenter Differences inazimuthal orientation of grains may introduce small scatter in ourresults which is on the order of 5
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ashington University Law
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Low-temperature rheology of calcite 131
Figure 2 Load-displacement curves for high quality (blue) and poor quality(red) indentations from the same grain at the same temperature Typical loaddisplacement curves have continuous curvature and occasionally show fewsmall increases in displacement (sim1ndash10 nm) referred to as pop ins A smallchange in displacement at the maximum load results from creep during the2 s hold in the loading function Load functions that deviate from this shapeoften imply that the data are not useful and are not used in this study
22 Nanoindentation
Load-controlled nanoindentation experiments in this study wereperformed using a diamond Berkovich indenter probe in a HysitronTI 950 TriboIndenter equipped with a temperature control stageIndividual indentation experiments consist of three segments 5 sof loading at a constant rate to the maximum load of 5 mN a 2-shold at the maximum load and 5 s of unloading at a constant rateWe use a maximum load of 5 mN to ensure contact depths aresufficiently deep to avoid complications with misshapen or bluntedindenter tips The indentation load P and probe displacement hare recorded simultaneously during each indentation experiment(Fig 2) Load measurements have a nominal precision of lt1 nNand displacement measurements have a precision of lt002 nm
The probe displacement is then converted to contact depth hc foreach indent using the maximum load applied Pmax and the initialunloading contact stiffness S which is determined from the slopeof the initial portion of the unloading curve
hc = h minus 075Pmax
S(1)
Contact area is related to contact depth through a polynomialfunction of the form
Ac = a0h2c + a1hc + a2h12
c + a3h14c + (2)
where ai are constants determined by indenting a material withwell-known indentation hardness and reduced elastic modulus overa wide range of loads Fused silica was indented over the range of100ndash10 000 μN for this calibration Indentation hardness H andreduced elastic modulus Er are calculated for each indent usingeqs (3) and (4) respectively and compared to the known values todetermine the accuracy of the area function
H = Pmax
Ac(3)
Er = Sradic
π
2radic
Ac
(4)
The area function is considered acceptable if the calculated valuesare within 10 of the known values
The measured load and displacement along with the contact areaare used to calculate the indentation hardness and the reduced elasticmodulus of the indented material The reduced elastic modulus isrelated to Youngrsquos modulus by
1
Er=
(1 minus ν2
E
)sample
+(
1 minus ν2
E
)indenter
(5)
where E is Youngrsquos modulus and ν is Poissonrsquos ratio and the sub-scripts refer to the properties of the sample and the indenter tip TheYoungrsquos modulus and Poissonrsquos ratio for the diamond indenter are1140 GPa and 007 respectively
The shape of the loadndashdisplacement curve is used to assess thequality of the data (Fig 2) Indents with atypical loadndashdisplacementcurves may indicate that the probe tip encountered something otherthan the flat sample surface (eg defects debris on the sample orsurface roughness) and are excluded from subsequent analysis
Between 20 and 25 indents were made in each grain at fourtemperatures (23 75 125 and 175 C) resulting in approximately100 indents in each grain and over 2000 indents in total The vol-ume affected by each indent extends beyond the contacted area andindividual indents were spaced 10 μm apart to avoid interactionbetween indents For experiments at all temperatures the samplewas clamped to the stage and allowed to equilibrate to the desiredtemperature for at least 30 min prior to the start of the experimentAfter the indents were completed the structure and morphologyof individual indents were examined in situ using Scanning ProbeMicroscopy (SPM) in the Hysitron TI 950 TriboIndenter as well asex situ using SEM SPM produces topographical images by scan-ning the nanoindenter probe across the sample while maintaininga constant load between the probe and the surface SPM imageswere taken of representative indentations in each orientation Usingthe SEM secondary electron images were taken of representativeindents produced at each temperature and orientation
23 Micropillar compression
Micropillars were fabricated using a focused ion beam (FIB) mi-cromilling technique that was originally developed and applied tometallic materials (Uchic et al 2009) Two instruments were used aZeiss Crossbeam 540 FIB-SEM and an FEI Quanta 3D FEG Priorto fabricating micropillars the entire Carrara marble sample wascoated with iridium to avoid sample charging The Ga+ ion beamwas then used to sputter away small amounts of the sample materialin concentric annular rings to produce a well with a cylindrical pil-lar at the centre To minimize ion implantation damage the pillarsare machined using progressively smaller concentric rings with pro-gressively lower ion beam doses to produce a 25 μm diameter wellwith a 1 or 3 μm diameter pillar in the centre The micropillars havean approximate diameter to height ratio of 12 Multiple micropil-lars were made in several grains oriented for compression parallelto 〈2243〉 (unfavorably oriented for twinning) and for compressionparallel to 〈1010〉 and 〈1120〉 (favorably oriented for twinning)
Micropillar compression experiments were conducted in loadcontrol using monotonic stair-step loading functions with alternat-ing segments of loading and holding Experiments were stopped
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132 MK Sly et al
when a sharp change in displacement with respect to time was ob-served which was assumed to coincide with the onset of brittlefailure or plastic yielding The load at which yielding occurred isconverted to yield stress by
σy = Py
Ap (6)
where Py is the load at the time of failure or yielding and Ap is thearea of the top of the micropillar prior to testing
24 Scanning transmission electron microscopy
A thin foil from a deformed micropillar was prepared using a Hi-tachi NB5000 focused ion and electron beam system The top of themicropillar was coated with a 1-μm-thick layer of carbon followedby a 1-μm-thick layer of platinum Additionally a 2-μm-thick plat-inum coating was deposited to protect both sides of the micropillarfrom collapsing during FIB lift-out A 20 kV beam with 07 nAcurrent was used to cut the foil from the base of the micropillarwhile beam currents of 007 nA at 10 kV and 001 nA at 5 kVwere used to perform rough and fine milling respectively to makethe foil electron transparent The resulting foil was mounted ona Cu grid for scanning transmission electron microscopy (STEM)experiments
STEM imaging was carried out using the aberration-correctedNion UltraSTEMTM 200 (operating at 200 and 100 kV) at OakRidge National Laboratory which is equipped with a fifth-orderaberration corrector and a cold-field emission electron gun TheCu grid was baked at 160 C under vacuum prior to the STEMexperiments to remove organics
3 R E S U LT S
31 Nanoindentation
The indentation hardness of calcite varies systematically with crys-tallographic orientation (Table 1 and Fig 3) Grains indented parallel([0001]) and subparallel (〈2243〉) to the c-axis are harder than thoseindented normal to the c-axis (〈1120〉 and 〈1010〉) The fifth ori-entation tested (〈4041〉) which lies between the c-axis [0001] and〈1010〉 exhibits intermediate hardness Differences between grainsof similar orientations are assumed to be due to small misalignmentsof grains with respect to the indenter
At room temperature (23 C) grains with hard orientations par-allel or subparallel to the c-axis have an average hardness of282 plusmn 012ndash286 plusmn 013 GPa Grains in softer orientations have anaverage hardness of 259 plusmn 010ndash260 plusmn 009 GPa Anisotropy ofhardness AH is defined as
AH =(
Hmax minus Hmin
Hall
)times 100 (7)
where Hmax is the average hardness of the grains in the strongestorientation Hmin is the average hardness of the grains in the weak-est orientation and Hall is the average hardness computed for allindents at a given temperature Hmax is always the average hardnessof the grains indented parallel or subparallel to the c-axis ([0001]or 〈2243〉) Hmin is always the average hardness of the grains in-dented in one of the two orientations normal to the c-axis (〈1120〉 or〈1010〉) At room temperature the anisotropy of hardness is 100
Indentation hardness of calcite is also temperature dependent(Table 1 and Fig 3) For orientations [0001] and 〈2243〉 the hardness
decreases to 216 plusmn 008 and 217 plusmn 005 GPa respectively at 175C Similarly the hardness in the softer orientations 〈1120〉 and〈1010〉 decreases to 198 plusmn 011 and 198 plusmn 008 GPa respectivelyat 175 C The anisotropy of hardness at 175 C is similar to that atroom temperature with a value of 92
Fig 4 shows representative secondary electron images of the tri-angular indents produced by the three-sided pyramidal Berkovichprobe Indent size and depth both increase with temperature Theapothem and depth of room temperature indents are approximately500 nm and 240ndash280 nm respectively The indents increase insize to an apothem of 750 nm and depths of 280ndash330 nm at175 C At lower temperatures the sides of indents performedin grains oriented with the indentation axis parallel or subparal-lel to the c-axis are rough and cracks are present The most extremecracking appears in grains indented parallel to the c-axis at roomtemperature (Fig 4) As temperature increases the number anddensity of cracks decreases and the sides of the indents becomesmoother There are only a few cracks in indents performed at125 C and no cracks observed in indents performed at 175 CThere is positive relief referred to as lsquopile-uprsquo around the sidesof the indents formed by material that is displaced during inden-tation SPM images in Fig 5 show additional examples of thisfeature
32 Micropillar compression
Fourteen micropillar compression experiments performed on pil-lars with a nominal diameter of 1 μm and oriented unfavorablyfor twinning (compression parallel to 〈2243〉) exhibit yield stressesvarying from 082 to 115 GPa at 23 C and 055ndash077 GPa at 175C Pillars were examined before and after experiments to iden-tify the mode of failure (Table 2) All pillars were characterizedas having failed by brittle fracture diffuse ductile deformation orvia deformation along shear bands Brittle fracture was identifiedby the presence of open cracks Diffuse ductile deformation wasidentified by a change in the shape of the pillar without any associ-ated cracking or deformation along shear bands Shear bands wereidentified in secondary electron images as narrow bands of highcontrast
Twelve 1 μm diameter pillars tested in grains oriented favorablyfor twinning (compression parallel to 〈1010〉) were more variablewith yield stresses ranging from 056 to 103 GPa at 23 C and 027ndash072 GPa at 175 C All of the 1 μm pillars show either fractureor diffuse ductile deformation without any noticeable shear bands(Table 2)
Additional micropillar compression experiments were conductedon 12 pillars with a diameter of 3 μm in both unfavorable (com-pression parallel to 〈2243〉) and favourable (compression parallelto 〈1010〉 and 〈1120〉)orientations for twinning The yield stressesof the 6 pillars in the unfavorable orientation are comparable tothose of the 1 μm pillars in the same orientation ranging from073 to 115 GPa However the 6 pillars tested in favourable ori-entations for twinning are markedly weaker than the 1 μm pillarstested under similar conditions with yield stresses ranging from010 to 036 GPa The 3 μm pillars exhibit notably different de-formation features from the 1 μm diameter pillars While brittleand diffuse plastic deformation are observed in the 3 μm pillarsthe larger samples also exhibit shear bands in the pillars favorablyoriented for twinning that are not apparent in pillars unfavorably ori-ented for twinning (Fig 6) STEM observation of these shear bandsindicates that they are deformation twins on e1018 Fig 7(a)
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Low-temperature rheology of calcite 133
Table 1 Indentation hardness as a function of orientation and temperature
Figure 3 Indentation hardness plotted against temperature Small trianglesrepresent the individual indentations (n = 2183) and large triangles rep-resent averages for 20ndash25 indentations performed in a single grain (colourcorresponds to the orientation of the grain following the colour scheme inFig 1) Error bars in the bottom of the figure which are offset for clarityshow plusmn one standard deviation for each orientation
shows a medium-angle annular dark field (MAADF) image of pil-lar 2086 P02 which is oriented nominally with the m1010planeas its surface with a shear band traversing across the pillar from top-left to bottom-right at an angle of sim59 from the horizontal Thisangle is comparable to the known angle between m and e of sim64Fig 7(b) shows a high-angle annular dark field (HAADF) image ofthe pillar with shear band highlighted as Region II and either sidesof the shear band highlighted as Regions I and III Fig 7(c) showsthe fast Fourier transform (FFT) diffraction patterns of Regions IndashIII We have used the crystal structure of calcite from Chessin et al(1965) to index the FFT patterns The FFT patterns confirm twin-ning during deformation with an in-plane rotation of 4378 for theshear band (Region II) with respect to the undeformed crystal oneither side (Region I and II) This angle is close to the theoreticalangle of 525 (Barber amp Wenk 1979) The differences between theangles measured here and the theoretical angles are likely becausethe STEM foil was not cut perfectly orthogonal to the twin Theout of plane direction corresponds to [2201] orientation of calcitecrystal structure (Chessin et al 1965)
4 D I S C U S S I O N
41 Yield stress from nanoindentation experiments
While indentation hardness measurements are precise and relativelystraightforward to collect there remains a significant challenge toscale these values to uniaxial properties used in traditional consti-tutive flow laws and models of deformation To relate indentationhardness measurements to uniaxial mechanical properties indenta-tion hardness must be converted to yield stress σy This relationshipis commonly expressed by a simple linear function where hardnessis proportional to yield stress via the constraint factor C (eq 8)(Tabor 1970)
H = Cσy (8)
The constraint factor may vary from 11 to 30 where 11 repre-sents the elastic limit and 30 represents the plastic limit (Johnson1970 Evans amp Goetze 1979) Constraint factors are typically around3 for materials with high Eσy ratios (gt133) such as metals and canbe as low as 15 for materials with low Eσy ratios such as glassesand polymers (Swain amp Hagan 1976 Fischer-Cripps 2011 Shawamp DeSalvo 2012) Calcite has a relatively low Eσy ratio whichmeans that the constraint factor is predicted to be in the range ofC = 15ndash30
Multiple models have been used to estimate the constraint factorand yield stress Here we present calculations for four such modelsJohnson (1970) Evans amp Goetze (1979) Mata et al (2002) Mataamp Alcala (2003) and Ginder et al (2018) (Table 3) As we haveno independent constraint on the yield stress at these deformationconditions we make no determination that one model is more validthan the others
The first of these models derived by Johnson (1970) comparesindentation to an expanding cavity in an elasticndashplastic solid Thismethod states that the ratio of indentation hardness to yield stress isdetermined by the geometry of the indenter and the elastic modulusof the material
H
σy= 2
3
[1 + ln
( E tan(θ)σy
+ 4 (1 minus 2ν)
6 (1 minus ν)
)] (9)
Here H is indentation hardness σy is yield stress E is Youngrsquosmodulus ν is Poissonrsquos ratio and θ is the angle between the surfaceof a conical indenter and the indented surface The value for θ
can be adjusted to compensate for different indenter tip geometriesby relating the volume of material displaced by differently shapedindenter probes (Fischer-Cripps 2011) We use eq (9) to determineyield stress though it can be simplified to eq (10) if a Poissonrsquosratio of 05 (ie an incompressible material) is assumed
H
σy= 2
3
[1 + ln
(1
3
E tan (θ )
σy
)] (10)
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ashington University Law
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134 MK Sly et al
Figure 4 Array of representative secondary electron images of indentations at each temperature step (rows) and in each orientation (columns) Images arecolored based on orientation following the colour scheme in Fig 1 The size of the indentations increases with increasing temperature apothems increase from500 to 750 nm and depths increase from 240 to 330 nm from 23 to 175 C Linear cracks are present in each of the four images of indentations performedparallel to 〈4041〉 (marked by dashed arrows) All of these features appear in the same orientation but do not appear to align with twin planes The imagesof these indentations were taken after the sample was coated with iridium in preparation for micropillar fabrication and these features are interpreted to becracks or tears in the iridium coating caused by the SEM electron beam There are bright linear features present in the lower right-hand face of the indentationsperformed parallel to 〈1010〉 and dark linear features present in the upper face of the indentations performed parallel to 〈1120〉 though these are less prominent(marked by solid arrows) Due to the orientation of these grains and the fact that these features are not present in other grains it is thought that these featuresare twin planes
Figure 5 Scanning probe microscope (SPM) images of indentations madeat room temperature in each orientation tested Images are colored basedon orientation following the colour scheme in Fig 1 Pile up is observed in4 of the 5 orientations (solid arrows) in varying amounts and on differentsides of the indentation This variation is due to the anisotropy of the calciteFractures are observed on the surface of the indentation parallel to [0001](dashed arrow)
Evans amp Goetze (1979) determine an empirical model based onthe expanding cavity model of Johnson (1970) using microindenta-tion data in olivine This model shows the same relationship betweenyield stress and indentation hardness as Johnsonrsquos model though
with slightly different coefficients
H
σy= 019 + 16 log
(E tan (θ )
σy
) (11)
Eq (11) is used in combination with Taborrsquos eq (8) to solve foryield stress and constraint factor simultaneously
Mata et al (2002) and Mata amp Alcala (2003) use finite elementanalysis to develop a new model for indentation in metals Mataamp Alcala use Johnsonrsquos idea that the constraint factor is a func-tion of the term ln(Eσr ) where σr is the reference stress at 10strain They then use finite element analysis to determine a polyno-mial function of ln(Eσr ) which is used to solve for the referencestress
H
σr=
sum4
i=0ci
[ln
(E
σr
)]i
(12)
Here ci are constants determined by the finite element analysisThis reference stress σr is comparable to the yield stresses deter-mined by Johnson (1970) Evans amp Goetze (1979) and Ginderet al (2018) The reference stress is then used in conjunction withthe curvature of the indentation load-displacement curve to deter-mine a pile-up factor and work hardening exponent that are usedto calculate yield stress Due to these additional steps the yieldstress determined by Mata amp Alcala is approximately a factor of4 lower than the reference stress However because the referencestress from Mata amp Alcala (2003) is defined the same way as theyield stresses from the other models we use it in subsequent com-parisons This model was initially developed for materials with yieldstrengths between 50 and 1000 MPa and Youngrsquos moduli between70 and 200 GPa The results from our study show that yield stress for
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ashington University Law
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Low-temperature rheology of calcite 135
Table 2 Micropillar compression results
calcite as defined in Mata amp Alcala (2003) (sim250 Mpa) is withinthe validated range and that the Youngrsquos modulus (50ndash65 GPa de-pending on temperature and orientation) is just below the validatedrange
Finally Ginder et al (2018) developed a model similar to thatof Johnson (1970) but for a power-law creeping solid as opposedto an elasticndashplastic solid This method uses Johnsonrsquos analysis ofthe dependence of indentation hardness on yield stress and Youngrsquos
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ashington University Law
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136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
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ashington University Law
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Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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ashington University Law
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Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
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140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 131
Figure 2 Load-displacement curves for high quality (blue) and poor quality(red) indentations from the same grain at the same temperature Typical loaddisplacement curves have continuous curvature and occasionally show fewsmall increases in displacement (sim1ndash10 nm) referred to as pop ins A smallchange in displacement at the maximum load results from creep during the2 s hold in the loading function Load functions that deviate from this shapeoften imply that the data are not useful and are not used in this study
22 Nanoindentation
Load-controlled nanoindentation experiments in this study wereperformed using a diamond Berkovich indenter probe in a HysitronTI 950 TriboIndenter equipped with a temperature control stageIndividual indentation experiments consist of three segments 5 sof loading at a constant rate to the maximum load of 5 mN a 2-shold at the maximum load and 5 s of unloading at a constant rateWe use a maximum load of 5 mN to ensure contact depths aresufficiently deep to avoid complications with misshapen or bluntedindenter tips The indentation load P and probe displacement hare recorded simultaneously during each indentation experiment(Fig 2) Load measurements have a nominal precision of lt1 nNand displacement measurements have a precision of lt002 nm
The probe displacement is then converted to contact depth hc foreach indent using the maximum load applied Pmax and the initialunloading contact stiffness S which is determined from the slopeof the initial portion of the unloading curve
hc = h minus 075Pmax
S(1)
Contact area is related to contact depth through a polynomialfunction of the form
Ac = a0h2c + a1hc + a2h12
c + a3h14c + (2)
where ai are constants determined by indenting a material withwell-known indentation hardness and reduced elastic modulus overa wide range of loads Fused silica was indented over the range of100ndash10 000 μN for this calibration Indentation hardness H andreduced elastic modulus Er are calculated for each indent usingeqs (3) and (4) respectively and compared to the known values todetermine the accuracy of the area function
H = Pmax
Ac(3)
Er = Sradic
π
2radic
Ac
(4)
The area function is considered acceptable if the calculated valuesare within 10 of the known values
The measured load and displacement along with the contact areaare used to calculate the indentation hardness and the reduced elasticmodulus of the indented material The reduced elastic modulus isrelated to Youngrsquos modulus by
1
Er=
(1 minus ν2
E
)sample
+(
1 minus ν2
E
)indenter
(5)
where E is Youngrsquos modulus and ν is Poissonrsquos ratio and the sub-scripts refer to the properties of the sample and the indenter tip TheYoungrsquos modulus and Poissonrsquos ratio for the diamond indenter are1140 GPa and 007 respectively
The shape of the loadndashdisplacement curve is used to assess thequality of the data (Fig 2) Indents with atypical loadndashdisplacementcurves may indicate that the probe tip encountered something otherthan the flat sample surface (eg defects debris on the sample orsurface roughness) and are excluded from subsequent analysis
Between 20 and 25 indents were made in each grain at fourtemperatures (23 75 125 and 175 C) resulting in approximately100 indents in each grain and over 2000 indents in total The vol-ume affected by each indent extends beyond the contacted area andindividual indents were spaced 10 μm apart to avoid interactionbetween indents For experiments at all temperatures the samplewas clamped to the stage and allowed to equilibrate to the desiredtemperature for at least 30 min prior to the start of the experimentAfter the indents were completed the structure and morphologyof individual indents were examined in situ using Scanning ProbeMicroscopy (SPM) in the Hysitron TI 950 TriboIndenter as well asex situ using SEM SPM produces topographical images by scan-ning the nanoindenter probe across the sample while maintaininga constant load between the probe and the surface SPM imageswere taken of representative indentations in each orientation Usingthe SEM secondary electron images were taken of representativeindents produced at each temperature and orientation
23 Micropillar compression
Micropillars were fabricated using a focused ion beam (FIB) mi-cromilling technique that was originally developed and applied tometallic materials (Uchic et al 2009) Two instruments were used aZeiss Crossbeam 540 FIB-SEM and an FEI Quanta 3D FEG Priorto fabricating micropillars the entire Carrara marble sample wascoated with iridium to avoid sample charging The Ga+ ion beamwas then used to sputter away small amounts of the sample materialin concentric annular rings to produce a well with a cylindrical pil-lar at the centre To minimize ion implantation damage the pillarsare machined using progressively smaller concentric rings with pro-gressively lower ion beam doses to produce a 25 μm diameter wellwith a 1 or 3 μm diameter pillar in the centre The micropillars havean approximate diameter to height ratio of 12 Multiple micropil-lars were made in several grains oriented for compression parallelto 〈2243〉 (unfavorably oriented for twinning) and for compressionparallel to 〈1010〉 and 〈1120〉 (favorably oriented for twinning)
Micropillar compression experiments were conducted in loadcontrol using monotonic stair-step loading functions with alternat-ing segments of loading and holding Experiments were stopped
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ashington University Law
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132 MK Sly et al
when a sharp change in displacement with respect to time was ob-served which was assumed to coincide with the onset of brittlefailure or plastic yielding The load at which yielding occurred isconverted to yield stress by
σy = Py
Ap (6)
where Py is the load at the time of failure or yielding and Ap is thearea of the top of the micropillar prior to testing
24 Scanning transmission electron microscopy
A thin foil from a deformed micropillar was prepared using a Hi-tachi NB5000 focused ion and electron beam system The top of themicropillar was coated with a 1-μm-thick layer of carbon followedby a 1-μm-thick layer of platinum Additionally a 2-μm-thick plat-inum coating was deposited to protect both sides of the micropillarfrom collapsing during FIB lift-out A 20 kV beam with 07 nAcurrent was used to cut the foil from the base of the micropillarwhile beam currents of 007 nA at 10 kV and 001 nA at 5 kVwere used to perform rough and fine milling respectively to makethe foil electron transparent The resulting foil was mounted ona Cu grid for scanning transmission electron microscopy (STEM)experiments
STEM imaging was carried out using the aberration-correctedNion UltraSTEMTM 200 (operating at 200 and 100 kV) at OakRidge National Laboratory which is equipped with a fifth-orderaberration corrector and a cold-field emission electron gun TheCu grid was baked at 160 C under vacuum prior to the STEMexperiments to remove organics
3 R E S U LT S
31 Nanoindentation
The indentation hardness of calcite varies systematically with crys-tallographic orientation (Table 1 and Fig 3) Grains indented parallel([0001]) and subparallel (〈2243〉) to the c-axis are harder than thoseindented normal to the c-axis (〈1120〉 and 〈1010〉) The fifth ori-entation tested (〈4041〉) which lies between the c-axis [0001] and〈1010〉 exhibits intermediate hardness Differences between grainsof similar orientations are assumed to be due to small misalignmentsof grains with respect to the indenter
At room temperature (23 C) grains with hard orientations par-allel or subparallel to the c-axis have an average hardness of282 plusmn 012ndash286 plusmn 013 GPa Grains in softer orientations have anaverage hardness of 259 plusmn 010ndash260 plusmn 009 GPa Anisotropy ofhardness AH is defined as
AH =(
Hmax minus Hmin
Hall
)times 100 (7)
where Hmax is the average hardness of the grains in the strongestorientation Hmin is the average hardness of the grains in the weak-est orientation and Hall is the average hardness computed for allindents at a given temperature Hmax is always the average hardnessof the grains indented parallel or subparallel to the c-axis ([0001]or 〈2243〉) Hmin is always the average hardness of the grains in-dented in one of the two orientations normal to the c-axis (〈1120〉 or〈1010〉) At room temperature the anisotropy of hardness is 100
Indentation hardness of calcite is also temperature dependent(Table 1 and Fig 3) For orientations [0001] and 〈2243〉 the hardness
decreases to 216 plusmn 008 and 217 plusmn 005 GPa respectively at 175C Similarly the hardness in the softer orientations 〈1120〉 and〈1010〉 decreases to 198 plusmn 011 and 198 plusmn 008 GPa respectivelyat 175 C The anisotropy of hardness at 175 C is similar to that atroom temperature with a value of 92
Fig 4 shows representative secondary electron images of the tri-angular indents produced by the three-sided pyramidal Berkovichprobe Indent size and depth both increase with temperature Theapothem and depth of room temperature indents are approximately500 nm and 240ndash280 nm respectively The indents increase insize to an apothem of 750 nm and depths of 280ndash330 nm at175 C At lower temperatures the sides of indents performedin grains oriented with the indentation axis parallel or subparal-lel to the c-axis are rough and cracks are present The most extremecracking appears in grains indented parallel to the c-axis at roomtemperature (Fig 4) As temperature increases the number anddensity of cracks decreases and the sides of the indents becomesmoother There are only a few cracks in indents performed at125 C and no cracks observed in indents performed at 175 CThere is positive relief referred to as lsquopile-uprsquo around the sidesof the indents formed by material that is displaced during inden-tation SPM images in Fig 5 show additional examples of thisfeature
32 Micropillar compression
Fourteen micropillar compression experiments performed on pil-lars with a nominal diameter of 1 μm and oriented unfavorablyfor twinning (compression parallel to 〈2243〉) exhibit yield stressesvarying from 082 to 115 GPa at 23 C and 055ndash077 GPa at 175C Pillars were examined before and after experiments to iden-tify the mode of failure (Table 2) All pillars were characterizedas having failed by brittle fracture diffuse ductile deformation orvia deformation along shear bands Brittle fracture was identifiedby the presence of open cracks Diffuse ductile deformation wasidentified by a change in the shape of the pillar without any associ-ated cracking or deformation along shear bands Shear bands wereidentified in secondary electron images as narrow bands of highcontrast
Twelve 1 μm diameter pillars tested in grains oriented favorablyfor twinning (compression parallel to 〈1010〉) were more variablewith yield stresses ranging from 056 to 103 GPa at 23 C and 027ndash072 GPa at 175 C All of the 1 μm pillars show either fractureor diffuse ductile deformation without any noticeable shear bands(Table 2)
Additional micropillar compression experiments were conductedon 12 pillars with a diameter of 3 μm in both unfavorable (com-pression parallel to 〈2243〉) and favourable (compression parallelto 〈1010〉 and 〈1120〉)orientations for twinning The yield stressesof the 6 pillars in the unfavorable orientation are comparable tothose of the 1 μm pillars in the same orientation ranging from073 to 115 GPa However the 6 pillars tested in favourable ori-entations for twinning are markedly weaker than the 1 μm pillarstested under similar conditions with yield stresses ranging from010 to 036 GPa The 3 μm pillars exhibit notably different de-formation features from the 1 μm diameter pillars While brittleand diffuse plastic deformation are observed in the 3 μm pillarsthe larger samples also exhibit shear bands in the pillars favorablyoriented for twinning that are not apparent in pillars unfavorably ori-ented for twinning (Fig 6) STEM observation of these shear bandsindicates that they are deformation twins on e1018 Fig 7(a)
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ashington University Law
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Low-temperature rheology of calcite 133
Table 1 Indentation hardness as a function of orientation and temperature
Figure 3 Indentation hardness plotted against temperature Small trianglesrepresent the individual indentations (n = 2183) and large triangles rep-resent averages for 20ndash25 indentations performed in a single grain (colourcorresponds to the orientation of the grain following the colour scheme inFig 1) Error bars in the bottom of the figure which are offset for clarityshow plusmn one standard deviation for each orientation
shows a medium-angle annular dark field (MAADF) image of pil-lar 2086 P02 which is oriented nominally with the m1010planeas its surface with a shear band traversing across the pillar from top-left to bottom-right at an angle of sim59 from the horizontal Thisangle is comparable to the known angle between m and e of sim64Fig 7(b) shows a high-angle annular dark field (HAADF) image ofthe pillar with shear band highlighted as Region II and either sidesof the shear band highlighted as Regions I and III Fig 7(c) showsthe fast Fourier transform (FFT) diffraction patterns of Regions IndashIII We have used the crystal structure of calcite from Chessin et al(1965) to index the FFT patterns The FFT patterns confirm twin-ning during deformation with an in-plane rotation of 4378 for theshear band (Region II) with respect to the undeformed crystal oneither side (Region I and II) This angle is close to the theoreticalangle of 525 (Barber amp Wenk 1979) The differences between theangles measured here and the theoretical angles are likely becausethe STEM foil was not cut perfectly orthogonal to the twin Theout of plane direction corresponds to [2201] orientation of calcitecrystal structure (Chessin et al 1965)
4 D I S C U S S I O N
41 Yield stress from nanoindentation experiments
While indentation hardness measurements are precise and relativelystraightforward to collect there remains a significant challenge toscale these values to uniaxial properties used in traditional consti-tutive flow laws and models of deformation To relate indentationhardness measurements to uniaxial mechanical properties indenta-tion hardness must be converted to yield stress σy This relationshipis commonly expressed by a simple linear function where hardnessis proportional to yield stress via the constraint factor C (eq 8)(Tabor 1970)
H = Cσy (8)
The constraint factor may vary from 11 to 30 where 11 repre-sents the elastic limit and 30 represents the plastic limit (Johnson1970 Evans amp Goetze 1979) Constraint factors are typically around3 for materials with high Eσy ratios (gt133) such as metals and canbe as low as 15 for materials with low Eσy ratios such as glassesand polymers (Swain amp Hagan 1976 Fischer-Cripps 2011 Shawamp DeSalvo 2012) Calcite has a relatively low Eσy ratio whichmeans that the constraint factor is predicted to be in the range ofC = 15ndash30
Multiple models have been used to estimate the constraint factorand yield stress Here we present calculations for four such modelsJohnson (1970) Evans amp Goetze (1979) Mata et al (2002) Mataamp Alcala (2003) and Ginder et al (2018) (Table 3) As we haveno independent constraint on the yield stress at these deformationconditions we make no determination that one model is more validthan the others
The first of these models derived by Johnson (1970) comparesindentation to an expanding cavity in an elasticndashplastic solid Thismethod states that the ratio of indentation hardness to yield stress isdetermined by the geometry of the indenter and the elastic modulusof the material
H
σy= 2
3
[1 + ln
( E tan(θ)σy
+ 4 (1 minus 2ν)
6 (1 minus ν)
)] (9)
Here H is indentation hardness σy is yield stress E is Youngrsquosmodulus ν is Poissonrsquos ratio and θ is the angle between the surfaceof a conical indenter and the indented surface The value for θ
can be adjusted to compensate for different indenter tip geometriesby relating the volume of material displaced by differently shapedindenter probes (Fischer-Cripps 2011) We use eq (9) to determineyield stress though it can be simplified to eq (10) if a Poissonrsquosratio of 05 (ie an incompressible material) is assumed
H
σy= 2
3
[1 + ln
(1
3
E tan (θ )
σy
)] (10)
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ashington University Law
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134 MK Sly et al
Figure 4 Array of representative secondary electron images of indentations at each temperature step (rows) and in each orientation (columns) Images arecolored based on orientation following the colour scheme in Fig 1 The size of the indentations increases with increasing temperature apothems increase from500 to 750 nm and depths increase from 240 to 330 nm from 23 to 175 C Linear cracks are present in each of the four images of indentations performedparallel to 〈4041〉 (marked by dashed arrows) All of these features appear in the same orientation but do not appear to align with twin planes The imagesof these indentations were taken after the sample was coated with iridium in preparation for micropillar fabrication and these features are interpreted to becracks or tears in the iridium coating caused by the SEM electron beam There are bright linear features present in the lower right-hand face of the indentationsperformed parallel to 〈1010〉 and dark linear features present in the upper face of the indentations performed parallel to 〈1120〉 though these are less prominent(marked by solid arrows) Due to the orientation of these grains and the fact that these features are not present in other grains it is thought that these featuresare twin planes
Figure 5 Scanning probe microscope (SPM) images of indentations madeat room temperature in each orientation tested Images are colored basedon orientation following the colour scheme in Fig 1 Pile up is observed in4 of the 5 orientations (solid arrows) in varying amounts and on differentsides of the indentation This variation is due to the anisotropy of the calciteFractures are observed on the surface of the indentation parallel to [0001](dashed arrow)
Evans amp Goetze (1979) determine an empirical model based onthe expanding cavity model of Johnson (1970) using microindenta-tion data in olivine This model shows the same relationship betweenyield stress and indentation hardness as Johnsonrsquos model though
with slightly different coefficients
H
σy= 019 + 16 log
(E tan (θ )
σy
) (11)
Eq (11) is used in combination with Taborrsquos eq (8) to solve foryield stress and constraint factor simultaneously
Mata et al (2002) and Mata amp Alcala (2003) use finite elementanalysis to develop a new model for indentation in metals Mataamp Alcala use Johnsonrsquos idea that the constraint factor is a func-tion of the term ln(Eσr ) where σr is the reference stress at 10strain They then use finite element analysis to determine a polyno-mial function of ln(Eσr ) which is used to solve for the referencestress
H
σr=
sum4
i=0ci
[ln
(E
σr
)]i
(12)
Here ci are constants determined by the finite element analysisThis reference stress σr is comparable to the yield stresses deter-mined by Johnson (1970) Evans amp Goetze (1979) and Ginderet al (2018) The reference stress is then used in conjunction withthe curvature of the indentation load-displacement curve to deter-mine a pile-up factor and work hardening exponent that are usedto calculate yield stress Due to these additional steps the yieldstress determined by Mata amp Alcala is approximately a factor of4 lower than the reference stress However because the referencestress from Mata amp Alcala (2003) is defined the same way as theyield stresses from the other models we use it in subsequent com-parisons This model was initially developed for materials with yieldstrengths between 50 and 1000 MPa and Youngrsquos moduli between70 and 200 GPa The results from our study show that yield stress for
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ashington University Law
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Low-temperature rheology of calcite 135
Table 2 Micropillar compression results
calcite as defined in Mata amp Alcala (2003) (sim250 Mpa) is withinthe validated range and that the Youngrsquos modulus (50ndash65 GPa de-pending on temperature and orientation) is just below the validatedrange
Finally Ginder et al (2018) developed a model similar to thatof Johnson (1970) but for a power-law creeping solid as opposedto an elasticndashplastic solid This method uses Johnsonrsquos analysis ofthe dependence of indentation hardness on yield stress and Youngrsquos
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ashington University Law
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136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
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ashington University Law
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Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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ashington University Law
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Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
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140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
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nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
132 MK Sly et al
when a sharp change in displacement with respect to time was ob-served which was assumed to coincide with the onset of brittlefailure or plastic yielding The load at which yielding occurred isconverted to yield stress by
σy = Py
Ap (6)
where Py is the load at the time of failure or yielding and Ap is thearea of the top of the micropillar prior to testing
24 Scanning transmission electron microscopy
A thin foil from a deformed micropillar was prepared using a Hi-tachi NB5000 focused ion and electron beam system The top of themicropillar was coated with a 1-μm-thick layer of carbon followedby a 1-μm-thick layer of platinum Additionally a 2-μm-thick plat-inum coating was deposited to protect both sides of the micropillarfrom collapsing during FIB lift-out A 20 kV beam with 07 nAcurrent was used to cut the foil from the base of the micropillarwhile beam currents of 007 nA at 10 kV and 001 nA at 5 kVwere used to perform rough and fine milling respectively to makethe foil electron transparent The resulting foil was mounted ona Cu grid for scanning transmission electron microscopy (STEM)experiments
STEM imaging was carried out using the aberration-correctedNion UltraSTEMTM 200 (operating at 200 and 100 kV) at OakRidge National Laboratory which is equipped with a fifth-orderaberration corrector and a cold-field emission electron gun TheCu grid was baked at 160 C under vacuum prior to the STEMexperiments to remove organics
3 R E S U LT S
31 Nanoindentation
The indentation hardness of calcite varies systematically with crys-tallographic orientation (Table 1 and Fig 3) Grains indented parallel([0001]) and subparallel (〈2243〉) to the c-axis are harder than thoseindented normal to the c-axis (〈1120〉 and 〈1010〉) The fifth ori-entation tested (〈4041〉) which lies between the c-axis [0001] and〈1010〉 exhibits intermediate hardness Differences between grainsof similar orientations are assumed to be due to small misalignmentsof grains with respect to the indenter
At room temperature (23 C) grains with hard orientations par-allel or subparallel to the c-axis have an average hardness of282 plusmn 012ndash286 plusmn 013 GPa Grains in softer orientations have anaverage hardness of 259 plusmn 010ndash260 plusmn 009 GPa Anisotropy ofhardness AH is defined as
AH =(
Hmax minus Hmin
Hall
)times 100 (7)
where Hmax is the average hardness of the grains in the strongestorientation Hmin is the average hardness of the grains in the weak-est orientation and Hall is the average hardness computed for allindents at a given temperature Hmax is always the average hardnessof the grains indented parallel or subparallel to the c-axis ([0001]or 〈2243〉) Hmin is always the average hardness of the grains in-dented in one of the two orientations normal to the c-axis (〈1120〉 or〈1010〉) At room temperature the anisotropy of hardness is 100
Indentation hardness of calcite is also temperature dependent(Table 1 and Fig 3) For orientations [0001] and 〈2243〉 the hardness
decreases to 216 plusmn 008 and 217 plusmn 005 GPa respectively at 175C Similarly the hardness in the softer orientations 〈1120〉 and〈1010〉 decreases to 198 plusmn 011 and 198 plusmn 008 GPa respectivelyat 175 C The anisotropy of hardness at 175 C is similar to that atroom temperature with a value of 92
Fig 4 shows representative secondary electron images of the tri-angular indents produced by the three-sided pyramidal Berkovichprobe Indent size and depth both increase with temperature Theapothem and depth of room temperature indents are approximately500 nm and 240ndash280 nm respectively The indents increase insize to an apothem of 750 nm and depths of 280ndash330 nm at175 C At lower temperatures the sides of indents performedin grains oriented with the indentation axis parallel or subparal-lel to the c-axis are rough and cracks are present The most extremecracking appears in grains indented parallel to the c-axis at roomtemperature (Fig 4) As temperature increases the number anddensity of cracks decreases and the sides of the indents becomesmoother There are only a few cracks in indents performed at125 C and no cracks observed in indents performed at 175 CThere is positive relief referred to as lsquopile-uprsquo around the sidesof the indents formed by material that is displaced during inden-tation SPM images in Fig 5 show additional examples of thisfeature
32 Micropillar compression
Fourteen micropillar compression experiments performed on pil-lars with a nominal diameter of 1 μm and oriented unfavorablyfor twinning (compression parallel to 〈2243〉) exhibit yield stressesvarying from 082 to 115 GPa at 23 C and 055ndash077 GPa at 175C Pillars were examined before and after experiments to iden-tify the mode of failure (Table 2) All pillars were characterizedas having failed by brittle fracture diffuse ductile deformation orvia deformation along shear bands Brittle fracture was identifiedby the presence of open cracks Diffuse ductile deformation wasidentified by a change in the shape of the pillar without any associ-ated cracking or deformation along shear bands Shear bands wereidentified in secondary electron images as narrow bands of highcontrast
Twelve 1 μm diameter pillars tested in grains oriented favorablyfor twinning (compression parallel to 〈1010〉) were more variablewith yield stresses ranging from 056 to 103 GPa at 23 C and 027ndash072 GPa at 175 C All of the 1 μm pillars show either fractureor diffuse ductile deformation without any noticeable shear bands(Table 2)
Additional micropillar compression experiments were conductedon 12 pillars with a diameter of 3 μm in both unfavorable (com-pression parallel to 〈2243〉) and favourable (compression parallelto 〈1010〉 and 〈1120〉)orientations for twinning The yield stressesof the 6 pillars in the unfavorable orientation are comparable tothose of the 1 μm pillars in the same orientation ranging from073 to 115 GPa However the 6 pillars tested in favourable ori-entations for twinning are markedly weaker than the 1 μm pillarstested under similar conditions with yield stresses ranging from010 to 036 GPa The 3 μm pillars exhibit notably different de-formation features from the 1 μm diameter pillars While brittleand diffuse plastic deformation are observed in the 3 μm pillarsthe larger samples also exhibit shear bands in the pillars favorablyoriented for twinning that are not apparent in pillars unfavorably ori-ented for twinning (Fig 6) STEM observation of these shear bandsindicates that they are deformation twins on e1018 Fig 7(a)
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Low-temperature rheology of calcite 133
Table 1 Indentation hardness as a function of orientation and temperature
Figure 3 Indentation hardness plotted against temperature Small trianglesrepresent the individual indentations (n = 2183) and large triangles rep-resent averages for 20ndash25 indentations performed in a single grain (colourcorresponds to the orientation of the grain following the colour scheme inFig 1) Error bars in the bottom of the figure which are offset for clarityshow plusmn one standard deviation for each orientation
shows a medium-angle annular dark field (MAADF) image of pil-lar 2086 P02 which is oriented nominally with the m1010planeas its surface with a shear band traversing across the pillar from top-left to bottom-right at an angle of sim59 from the horizontal Thisangle is comparable to the known angle between m and e of sim64Fig 7(b) shows a high-angle annular dark field (HAADF) image ofthe pillar with shear band highlighted as Region II and either sidesof the shear band highlighted as Regions I and III Fig 7(c) showsthe fast Fourier transform (FFT) diffraction patterns of Regions IndashIII We have used the crystal structure of calcite from Chessin et al(1965) to index the FFT patterns The FFT patterns confirm twin-ning during deformation with an in-plane rotation of 4378 for theshear band (Region II) with respect to the undeformed crystal oneither side (Region I and II) This angle is close to the theoreticalangle of 525 (Barber amp Wenk 1979) The differences between theangles measured here and the theoretical angles are likely becausethe STEM foil was not cut perfectly orthogonal to the twin Theout of plane direction corresponds to [2201] orientation of calcitecrystal structure (Chessin et al 1965)
4 D I S C U S S I O N
41 Yield stress from nanoindentation experiments
While indentation hardness measurements are precise and relativelystraightforward to collect there remains a significant challenge toscale these values to uniaxial properties used in traditional consti-tutive flow laws and models of deformation To relate indentationhardness measurements to uniaxial mechanical properties indenta-tion hardness must be converted to yield stress σy This relationshipis commonly expressed by a simple linear function where hardnessis proportional to yield stress via the constraint factor C (eq 8)(Tabor 1970)
H = Cσy (8)
The constraint factor may vary from 11 to 30 where 11 repre-sents the elastic limit and 30 represents the plastic limit (Johnson1970 Evans amp Goetze 1979) Constraint factors are typically around3 for materials with high Eσy ratios (gt133) such as metals and canbe as low as 15 for materials with low Eσy ratios such as glassesand polymers (Swain amp Hagan 1976 Fischer-Cripps 2011 Shawamp DeSalvo 2012) Calcite has a relatively low Eσy ratio whichmeans that the constraint factor is predicted to be in the range ofC = 15ndash30
Multiple models have been used to estimate the constraint factorand yield stress Here we present calculations for four such modelsJohnson (1970) Evans amp Goetze (1979) Mata et al (2002) Mataamp Alcala (2003) and Ginder et al (2018) (Table 3) As we haveno independent constraint on the yield stress at these deformationconditions we make no determination that one model is more validthan the others
The first of these models derived by Johnson (1970) comparesindentation to an expanding cavity in an elasticndashplastic solid Thismethod states that the ratio of indentation hardness to yield stress isdetermined by the geometry of the indenter and the elastic modulusof the material
H
σy= 2
3
[1 + ln
( E tan(θ)σy
+ 4 (1 minus 2ν)
6 (1 minus ν)
)] (9)
Here H is indentation hardness σy is yield stress E is Youngrsquosmodulus ν is Poissonrsquos ratio and θ is the angle between the surfaceof a conical indenter and the indented surface The value for θ
can be adjusted to compensate for different indenter tip geometriesby relating the volume of material displaced by differently shapedindenter probes (Fischer-Cripps 2011) We use eq (9) to determineyield stress though it can be simplified to eq (10) if a Poissonrsquosratio of 05 (ie an incompressible material) is assumed
H
σy= 2
3
[1 + ln
(1
3
E tan (θ )
σy
)] (10)
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134 MK Sly et al
Figure 4 Array of representative secondary electron images of indentations at each temperature step (rows) and in each orientation (columns) Images arecolored based on orientation following the colour scheme in Fig 1 The size of the indentations increases with increasing temperature apothems increase from500 to 750 nm and depths increase from 240 to 330 nm from 23 to 175 C Linear cracks are present in each of the four images of indentations performedparallel to 〈4041〉 (marked by dashed arrows) All of these features appear in the same orientation but do not appear to align with twin planes The imagesof these indentations were taken after the sample was coated with iridium in preparation for micropillar fabrication and these features are interpreted to becracks or tears in the iridium coating caused by the SEM electron beam There are bright linear features present in the lower right-hand face of the indentationsperformed parallel to 〈1010〉 and dark linear features present in the upper face of the indentations performed parallel to 〈1120〉 though these are less prominent(marked by solid arrows) Due to the orientation of these grains and the fact that these features are not present in other grains it is thought that these featuresare twin planes
Figure 5 Scanning probe microscope (SPM) images of indentations madeat room temperature in each orientation tested Images are colored basedon orientation following the colour scheme in Fig 1 Pile up is observed in4 of the 5 orientations (solid arrows) in varying amounts and on differentsides of the indentation This variation is due to the anisotropy of the calciteFractures are observed on the surface of the indentation parallel to [0001](dashed arrow)
Evans amp Goetze (1979) determine an empirical model based onthe expanding cavity model of Johnson (1970) using microindenta-tion data in olivine This model shows the same relationship betweenyield stress and indentation hardness as Johnsonrsquos model though
with slightly different coefficients
H
σy= 019 + 16 log
(E tan (θ )
σy
) (11)
Eq (11) is used in combination with Taborrsquos eq (8) to solve foryield stress and constraint factor simultaneously
Mata et al (2002) and Mata amp Alcala (2003) use finite elementanalysis to develop a new model for indentation in metals Mataamp Alcala use Johnsonrsquos idea that the constraint factor is a func-tion of the term ln(Eσr ) where σr is the reference stress at 10strain They then use finite element analysis to determine a polyno-mial function of ln(Eσr ) which is used to solve for the referencestress
H
σr=
sum4
i=0ci
[ln
(E
σr
)]i
(12)
Here ci are constants determined by the finite element analysisThis reference stress σr is comparable to the yield stresses deter-mined by Johnson (1970) Evans amp Goetze (1979) and Ginderet al (2018) The reference stress is then used in conjunction withthe curvature of the indentation load-displacement curve to deter-mine a pile-up factor and work hardening exponent that are usedto calculate yield stress Due to these additional steps the yieldstress determined by Mata amp Alcala is approximately a factor of4 lower than the reference stress However because the referencestress from Mata amp Alcala (2003) is defined the same way as theyield stresses from the other models we use it in subsequent com-parisons This model was initially developed for materials with yieldstrengths between 50 and 1000 MPa and Youngrsquos moduli between70 and 200 GPa The results from our study show that yield stress for
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ashington University Law
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Low-temperature rheology of calcite 135
Table 2 Micropillar compression results
calcite as defined in Mata amp Alcala (2003) (sim250 Mpa) is withinthe validated range and that the Youngrsquos modulus (50ndash65 GPa de-pending on temperature and orientation) is just below the validatedrange
Finally Ginder et al (2018) developed a model similar to thatof Johnson (1970) but for a power-law creeping solid as opposedto an elasticndashplastic solid This method uses Johnsonrsquos analysis ofthe dependence of indentation hardness on yield stress and Youngrsquos
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ashington University Law
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136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
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ashington University Law
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Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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ashington University Law
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Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
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ashington University Law
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140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
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nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
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ashington University Law
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Low-temperature rheology of calcite 133
Table 1 Indentation hardness as a function of orientation and temperature
Figure 3 Indentation hardness plotted against temperature Small trianglesrepresent the individual indentations (n = 2183) and large triangles rep-resent averages for 20ndash25 indentations performed in a single grain (colourcorresponds to the orientation of the grain following the colour scheme inFig 1) Error bars in the bottom of the figure which are offset for clarityshow plusmn one standard deviation for each orientation
shows a medium-angle annular dark field (MAADF) image of pil-lar 2086 P02 which is oriented nominally with the m1010planeas its surface with a shear band traversing across the pillar from top-left to bottom-right at an angle of sim59 from the horizontal Thisangle is comparable to the known angle between m and e of sim64Fig 7(b) shows a high-angle annular dark field (HAADF) image ofthe pillar with shear band highlighted as Region II and either sidesof the shear band highlighted as Regions I and III Fig 7(c) showsthe fast Fourier transform (FFT) diffraction patterns of Regions IndashIII We have used the crystal structure of calcite from Chessin et al(1965) to index the FFT patterns The FFT patterns confirm twin-ning during deformation with an in-plane rotation of 4378 for theshear band (Region II) with respect to the undeformed crystal oneither side (Region I and II) This angle is close to the theoreticalangle of 525 (Barber amp Wenk 1979) The differences between theangles measured here and the theoretical angles are likely becausethe STEM foil was not cut perfectly orthogonal to the twin Theout of plane direction corresponds to [2201] orientation of calcitecrystal structure (Chessin et al 1965)
4 D I S C U S S I O N
41 Yield stress from nanoindentation experiments
While indentation hardness measurements are precise and relativelystraightforward to collect there remains a significant challenge toscale these values to uniaxial properties used in traditional consti-tutive flow laws and models of deformation To relate indentationhardness measurements to uniaxial mechanical properties indenta-tion hardness must be converted to yield stress σy This relationshipis commonly expressed by a simple linear function where hardnessis proportional to yield stress via the constraint factor C (eq 8)(Tabor 1970)
H = Cσy (8)
The constraint factor may vary from 11 to 30 where 11 repre-sents the elastic limit and 30 represents the plastic limit (Johnson1970 Evans amp Goetze 1979) Constraint factors are typically around3 for materials with high Eσy ratios (gt133) such as metals and canbe as low as 15 for materials with low Eσy ratios such as glassesand polymers (Swain amp Hagan 1976 Fischer-Cripps 2011 Shawamp DeSalvo 2012) Calcite has a relatively low Eσy ratio whichmeans that the constraint factor is predicted to be in the range ofC = 15ndash30
Multiple models have been used to estimate the constraint factorand yield stress Here we present calculations for four such modelsJohnson (1970) Evans amp Goetze (1979) Mata et al (2002) Mataamp Alcala (2003) and Ginder et al (2018) (Table 3) As we haveno independent constraint on the yield stress at these deformationconditions we make no determination that one model is more validthan the others
The first of these models derived by Johnson (1970) comparesindentation to an expanding cavity in an elasticndashplastic solid Thismethod states that the ratio of indentation hardness to yield stress isdetermined by the geometry of the indenter and the elastic modulusof the material
H
σy= 2
3
[1 + ln
( E tan(θ)σy
+ 4 (1 minus 2ν)
6 (1 minus ν)
)] (9)
Here H is indentation hardness σy is yield stress E is Youngrsquosmodulus ν is Poissonrsquos ratio and θ is the angle between the surfaceof a conical indenter and the indented surface The value for θ
can be adjusted to compensate for different indenter tip geometriesby relating the volume of material displaced by differently shapedindenter probes (Fischer-Cripps 2011) We use eq (9) to determineyield stress though it can be simplified to eq (10) if a Poissonrsquosratio of 05 (ie an incompressible material) is assumed
H
σy= 2
3
[1 + ln
(1
3
E tan (θ )
σy
)] (10)
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ashington University Law
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134 MK Sly et al
Figure 4 Array of representative secondary electron images of indentations at each temperature step (rows) and in each orientation (columns) Images arecolored based on orientation following the colour scheme in Fig 1 The size of the indentations increases with increasing temperature apothems increase from500 to 750 nm and depths increase from 240 to 330 nm from 23 to 175 C Linear cracks are present in each of the four images of indentations performedparallel to 〈4041〉 (marked by dashed arrows) All of these features appear in the same orientation but do not appear to align with twin planes The imagesof these indentations were taken after the sample was coated with iridium in preparation for micropillar fabrication and these features are interpreted to becracks or tears in the iridium coating caused by the SEM electron beam There are bright linear features present in the lower right-hand face of the indentationsperformed parallel to 〈1010〉 and dark linear features present in the upper face of the indentations performed parallel to 〈1120〉 though these are less prominent(marked by solid arrows) Due to the orientation of these grains and the fact that these features are not present in other grains it is thought that these featuresare twin planes
Figure 5 Scanning probe microscope (SPM) images of indentations madeat room temperature in each orientation tested Images are colored basedon orientation following the colour scheme in Fig 1 Pile up is observed in4 of the 5 orientations (solid arrows) in varying amounts and on differentsides of the indentation This variation is due to the anisotropy of the calciteFractures are observed on the surface of the indentation parallel to [0001](dashed arrow)
Evans amp Goetze (1979) determine an empirical model based onthe expanding cavity model of Johnson (1970) using microindenta-tion data in olivine This model shows the same relationship betweenyield stress and indentation hardness as Johnsonrsquos model though
with slightly different coefficients
H
σy= 019 + 16 log
(E tan (θ )
σy
) (11)
Eq (11) is used in combination with Taborrsquos eq (8) to solve foryield stress and constraint factor simultaneously
Mata et al (2002) and Mata amp Alcala (2003) use finite elementanalysis to develop a new model for indentation in metals Mataamp Alcala use Johnsonrsquos idea that the constraint factor is a func-tion of the term ln(Eσr ) where σr is the reference stress at 10strain They then use finite element analysis to determine a polyno-mial function of ln(Eσr ) which is used to solve for the referencestress
H
σr=
sum4
i=0ci
[ln
(E
σr
)]i
(12)
Here ci are constants determined by the finite element analysisThis reference stress σr is comparable to the yield stresses deter-mined by Johnson (1970) Evans amp Goetze (1979) and Ginderet al (2018) The reference stress is then used in conjunction withthe curvature of the indentation load-displacement curve to deter-mine a pile-up factor and work hardening exponent that are usedto calculate yield stress Due to these additional steps the yieldstress determined by Mata amp Alcala is approximately a factor of4 lower than the reference stress However because the referencestress from Mata amp Alcala (2003) is defined the same way as theyield stresses from the other models we use it in subsequent com-parisons This model was initially developed for materials with yieldstrengths between 50 and 1000 MPa and Youngrsquos moduli between70 and 200 GPa The results from our study show that yield stress for
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ashington University Law
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Low-temperature rheology of calcite 135
Table 2 Micropillar compression results
calcite as defined in Mata amp Alcala (2003) (sim250 Mpa) is withinthe validated range and that the Youngrsquos modulus (50ndash65 GPa de-pending on temperature and orientation) is just below the validatedrange
Finally Ginder et al (2018) developed a model similar to thatof Johnson (1970) but for a power-law creeping solid as opposedto an elasticndashplastic solid This method uses Johnsonrsquos analysis ofthe dependence of indentation hardness on yield stress and Youngrsquos
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ashington University Law
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136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
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ashington University Law
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Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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ashington University Law
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Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
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ashington University Law
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140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
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nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
134 MK Sly et al
Figure 4 Array of representative secondary electron images of indentations at each temperature step (rows) and in each orientation (columns) Images arecolored based on orientation following the colour scheme in Fig 1 The size of the indentations increases with increasing temperature apothems increase from500 to 750 nm and depths increase from 240 to 330 nm from 23 to 175 C Linear cracks are present in each of the four images of indentations performedparallel to 〈4041〉 (marked by dashed arrows) All of these features appear in the same orientation but do not appear to align with twin planes The imagesof these indentations were taken after the sample was coated with iridium in preparation for micropillar fabrication and these features are interpreted to becracks or tears in the iridium coating caused by the SEM electron beam There are bright linear features present in the lower right-hand face of the indentationsperformed parallel to 〈1010〉 and dark linear features present in the upper face of the indentations performed parallel to 〈1120〉 though these are less prominent(marked by solid arrows) Due to the orientation of these grains and the fact that these features are not present in other grains it is thought that these featuresare twin planes
Figure 5 Scanning probe microscope (SPM) images of indentations madeat room temperature in each orientation tested Images are colored basedon orientation following the colour scheme in Fig 1 Pile up is observed in4 of the 5 orientations (solid arrows) in varying amounts and on differentsides of the indentation This variation is due to the anisotropy of the calciteFractures are observed on the surface of the indentation parallel to [0001](dashed arrow)
Evans amp Goetze (1979) determine an empirical model based onthe expanding cavity model of Johnson (1970) using microindenta-tion data in olivine This model shows the same relationship betweenyield stress and indentation hardness as Johnsonrsquos model though
with slightly different coefficients
H
σy= 019 + 16 log
(E tan (θ )
σy
) (11)
Eq (11) is used in combination with Taborrsquos eq (8) to solve foryield stress and constraint factor simultaneously
Mata et al (2002) and Mata amp Alcala (2003) use finite elementanalysis to develop a new model for indentation in metals Mataamp Alcala use Johnsonrsquos idea that the constraint factor is a func-tion of the term ln(Eσr ) where σr is the reference stress at 10strain They then use finite element analysis to determine a polyno-mial function of ln(Eσr ) which is used to solve for the referencestress
H
σr=
sum4
i=0ci
[ln
(E
σr
)]i
(12)
Here ci are constants determined by the finite element analysisThis reference stress σr is comparable to the yield stresses deter-mined by Johnson (1970) Evans amp Goetze (1979) and Ginderet al (2018) The reference stress is then used in conjunction withthe curvature of the indentation load-displacement curve to deter-mine a pile-up factor and work hardening exponent that are usedto calculate yield stress Due to these additional steps the yieldstress determined by Mata amp Alcala is approximately a factor of4 lower than the reference stress However because the referencestress from Mata amp Alcala (2003) is defined the same way as theyield stresses from the other models we use it in subsequent com-parisons This model was initially developed for materials with yieldstrengths between 50 and 1000 MPa and Youngrsquos moduli between70 and 200 GPa The results from our study show that yield stress for
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 135
Table 2 Micropillar compression results
calcite as defined in Mata amp Alcala (2003) (sim250 Mpa) is withinthe validated range and that the Youngrsquos modulus (50ndash65 GPa de-pending on temperature and orientation) is just below the validatedrange
Finally Ginder et al (2018) developed a model similar to thatof Johnson (1970) but for a power-law creeping solid as opposedto an elasticndashplastic solid This method uses Johnsonrsquos analysis ofthe dependence of indentation hardness on yield stress and Youngrsquos
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ashington University Law
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136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
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Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 135
Table 2 Micropillar compression results
calcite as defined in Mata amp Alcala (2003) (sim250 Mpa) is withinthe validated range and that the Youngrsquos modulus (50ndash65 GPa de-pending on temperature and orientation) is just below the validatedrange
Finally Ginder et al (2018) developed a model similar to thatof Johnson (1970) but for a power-law creeping solid as opposedto an elasticndashplastic solid This method uses Johnsonrsquos analysis ofthe dependence of indentation hardness on yield stress and Youngrsquos
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
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nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
136 MK Sly et al
Figure 6 3 μm diameter micropillars before (left-hand side) and after (right-hand side) a compression test (a) A pillar oriented unfavorably for twinning(experiment 1008 PO2) The image taken after deformation shows that the pillar changed shape uniformly exhibiting mainly diffuse ductile deformation withsome minor fractures forming near the top of the pillar (b) A pillar oriented favorably for twinning (experiment 1018 P00) There is a shear band shown inbright relief at approximately a 45 angle to the axis of the pillar that appears after testing
modulus during the indentation of an elasticndashplastic solid (Johnson1970) and simplifies the finite element model developed by Bowerthat takes a similar approach (Bower et al 1993) Like Bowerrsquosmodel Ginderrsquos method correlates two power-law relationshipsone for uniaxial compression ε = ασ n
y and one for indentationεi = β H n where ε is uniaxial strain rate σy is yield stress εi
is the indentation strain rate H is the indentation hardness andn is the stress exponent α and β are terms related to creep andindentation creep respectively and take an Arrhenius form Theseexpressions are related to one another by α = βFn where F isthe reduced contact pressure and a function of the stress exponentn derived from Johnsonrsquos interpretation of the cavity expansionproblem (Johnson 1970 Ginder et al 2018) An explicit assumptionin the Ginder model is that the rheology of the tested material isbest modelled as a power law and is suitable for stress exponentsless than sim7 However it is previously observed that the stressexponent of calcite is likely greater than 9 and may be as great as
70 even in experiments at high temperature (De Bresser amp Spiers1997) Higher stress exponents in the Ginder model results in lowercalculated yield stresses As such we use a stress exponent of 7 inour calculation which represents a conservative upper bound on theyield stress determined using the Ginder model
42 Yield stress from micropillar compression
The 3 μm diameter pillars show a greater variation in yield stresswith crystal orientation than the 1 μm diameter pillars The yieldstresses for the 1 μm pillars are distributed widely over the totalrange of yield stresses while the 3 μm pillars show a distinct con-trast between the pillars oriented favorably for twinning and thoseoriented unfavorably for twinning (Table 2) The differences areinterpreted to be a consequence of the size of the micropillars the3 μm pillars are approximately 30 times larger in volume than the1 μm pillars Smaller pillars appear to suppress the formation of
Dow
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
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Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
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icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
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Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 137
Figure 7 (a) A medium-angle annular dark field (MAADF) image of pillar 2086 P02 with a shear band traversing the pillar from the top left to bottom centreThe white box shows the area shown in (b) (b) A high-angle annular dark field (HAADF) image of the pillar rotated by 30 Scale bars in A and B correspondto 300 and 30 nm respectively Regions I and III are the undeformed regions of the material and region II is the twin (c) Fast-Fourier transform patternsof Regions I II and III as highlighted in B This shows an in-plane rotation of 44 between the twin red and the host green This angle differs from thetheoretical angle of rotation because the sample was not perfectly orthogonal to the twin plane
Table 3 Yield stress as a function of orientation and temperature of the four models used
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
138 MK Sly et al
0 200 400 600 800 1000 1200Temperature (K)
0
05
1
15
2
25
3
35
4
45
5
Yiel
d St
ress
(GPa
)
-200 0 200 400 600 800Temperature (degC)
p=1 q=1p=12 q=1p=1 q=2p=12 q=2Johnson (1970)Evans and Goetze (1979)
Ginder et al (2018)Mata and Alcala (2003)
This Study
Renner et al (2002)
Figure 8 Several combinations of p and q were chosen for fitting eq (13) for each of the four models described above Resulting flow laws are shown ascolored lines and fitting parameters are provided in Table 4 The shaded region shows the temperature range of our experiments The flow laws from our studyare in good agreement with higher temperature results from Renner et al (2002) shown in grey circles
twins and shear bands and are inferred to exhibit more stochas-tic behavior resulting in a wider range of yield stresses (Table 2)Similar stochastic behavior has been observed in metallic glasses(Bharathula et al 2010) The variation in yield stresses of the 1 μmpillars could also be due to differences in defect density from pillarto pillar or a preexisting twin in the micropillar The 3 μm pillarsmay have a more representative defect density making their behaviormore predictable The difference in yield stresses between favorablyand unfavorably oriented 3 μm pillars is interpreted to be due tothe ease of deformation twinning in the favorably oriented pillarsOther slip systems namely r1014〈2021〉 and f1012〈2201〉 re-quire higher stresses to activate than the twin system (De Bresseramp Spiers 1997) The yield stresses in pillars oriented favorably fortwinning are on average sim50 lower than the yield stresses calcu-lated from indentation in grains of similar orientations while thepillars oriented unfavorably for twinning show similar though stillslightly lower yield stresses to their indentation counterparts Thisis likely caused by the geometry of the sample and the number ofactive slip systems required for plastic deformation It is assumedthat secondary slip systems are activated during indentation exper-iments which are needed to accommodate the spherical pattern ofdeformation In contrast a micropillar may require only one slipsystem (Evans amp Goetze 1979) Hence we conclude that the yieldstresses determined for the twinned micropillars represent a lowerlimit on the rheology of calcite and are in general agreement withthe data obtained from nanoindentation experiments
43 Low-temperature flow laws
We fit our indentation data to a flow law for plasticity limited bylattice resistance [from eq 212 from Frost amp Ashby (1982)]
ε = Aσ 2y exp
minus H lowast
RT
[1 minus
(σy
σp
)p]q (13)
where ε is strain rate A is a pre-exponential factor Hlowast is the acti-vation enthalpy R is the gas constant T is temperature σy is yieldstress σp is athermal Peierls stress and p and q are dimensionlessquantities that depend on the energy barriers to dislocation motionwith values 0 lt p le 1 and 1 le q le 2 (Kocks et al 1975) Calculatedyield stresses and strain rates for several predefined values for p andq are used to fit the athermal Peierls stress σp and A to eq (13)(Table 2and Fig 8) A value of activation enthalpy Hlowast of 200 kJmolminus1 has been determined in a previous study and is used to fit thedata here (Renner et al 2002) Values for A of 64 times 105ndash20 times 1010
GPaminus2sminus1 and σp of 12ndash99 GPa were calculated using MATLABrsquosCurve Fitting Tool
44 Size effect
Additional indentation experiments were conducted over a range ofmaximum loads of 01ndash25 mN in five individual grains to investigatethe effect of contact radius on indentation hardness The maximumloads correspond to contact radii ranging from 02 to 18 μm Thesetests show a weak dependence of indentation hardness on contactarea of the form H prop ac
m with m = ndash005 (Fig 9) ac is the contactradius defined by ac = hc tan(θ ) where hc is the contact depth ofa given indent and θ is the effective cone angle of a Berkovich tip(Fischer-Cripps 2011) The majority of the curvature observed inthis study occurs at very small contact radii and indentation hardnessremains fairly constant for contact radii ge sim1 μm While the effectof indentation size on hardness is minor additional tests at higherloads or with larger indenter tips are necessary to fully understandthe indentation size effect in calcite
Given that the maximum load for all indents in this study wasset to a constant value of 5 mN the size effect does not affectthe results of our study For a given maximum load differencesin contact radius reflect differences in indentation hardness and byextension yield stress
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 139
Table 4 Flow law parameters
p = 1 q = 1 p = 12 q = 1 p = 1 q = 2 p = 12 q = 2
Johnson (1970)
A (Gpaminus2sminus1) 200 times 1010350times1010
500times109 200 times 1010387times1010
135times109 200 times 1010380times1010
203times109 813 times 108175times109
120times108
σ p (GPa) 284290186 488512
463 462476447 10731154
991
Evans amp Goetze (1979)
A (Gpaminus2sminus1) 200 times 1010335times1010
653times109 517 times 109939times109
960times108 200 times 1010363times1010
374times109 513 times 107988times107
381times106
σ p (GPa) 182186179 204207
201 299305288 564597
331
Mata amp Alcala (2003)
A (Gpaminus2sminus1) 200 times 1010313times1010
874times109 116 times 108185 times108
467 times107 200 times 1010347 times1010
529 times109 794 times 105132 times106
267 times 105
σ p (GPa) 160162157 209215
203 260267254 380396
364
Ginder et al (2018)
A (Gpaminus2sminus1) 200 times 1010309 times1010
907 times109 933 times 107147 times108
399 times107 200 times 1010346 times1010
544 times109 638 times 105104 times106
233times 105
σ p (GPa) 121123120 156160
152 197202192 280292
269
Values for the preexponential A and athermal Peierls stress σ p are shown for each set of p and q 95 confidence intervals are shown as super- and subscriptsrepresenting the positive and negative confidence intervals respectively
Figure 9 Indentation experiments done over a range of contact radii to look at the effect of indent size on indentation hardness Indentation experiments werebinned based on indentation contact depth into 20 nm bins then converted to contact radius The average (grey triangles) and standard deviation (error bars)for indentation hardness in each bin are shown From this we see that the majority of the size effect appears at very low contact radii but ac gt sim08 showslittle variation A power-law relationship is fit to the data (black line) and gives a relationship of H = 232aminus006
c
45 Comparison to previous studies
At temperatures greater than sim800 C the four different mod-els used to determine yield stress are in good agreement with oneanother (Fig 8) Moreover the yield stresses obtained through thesecalculations are broadly consistent with the results from triaxialdeformation experiments by Renner amp Evans (2002) when ex-trapolated to the temperature range and strain-rates used in thatstudy Rennerrsquos study was conducted using synthetic polycrystalcalcite samples with grain sizes ranging from sim6 to gt 65 μm atstrain rates of 5 times 10minus7ndash3 times 10minus3 sminus1 and temperatures of 600ndash800 C Extrapolating Rennerrsquos results to the higher strain-ratesof this study the yield stresses range from sim250 MPa at 600 Cdown to sim80 MPa at 800 C (Fig 8) The athermal Peierls stresses
calculated by Renner et al (2002) (asymp047ndash15 GPa depending ongrain size) are lower than those determined in this study The differ-ence is likely due to the different temperature ranges in each studymuch of the curvature of the flow law occurs at low temperatures(T lt sim500 K) which makes it difficult to constrain the athermalPeierls stress using only data from experiments at T gt 800 Further-more differences in the formulation of the low temperature plasticflow laws may explain some differences in the calculated flow lawparameters Renner et al (2002) define their flow law as
ε = Aσ 2y exp
(σy
σpminus Q
RT
) (14)
where Q is activation energy and σp includes a temperature depen-dent Peierls stress as well as a grain size dependent term (Renner amp
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
140 MK Sly et al
Evans 2002 Renner et al 2002) All other parameters are the sameas in eq (43) This formulation allows the yield stress to exceed theathermal Peierls stress given a finite strain rate at T = 0 K whereaseq (43) requires σy = σp at T = 0 K Although our flow laws agreewith the results from Renner et al (2002) at higher temperature (Tgt 800 K) it is also likely that different deformation mechanisms areactive in the two studies dislocation creep in the study by Renneret al (2002) and low temperature plasticity in our study and thatdirect comparison may not be appropriate
The results from our study are less consistent with the results fromstudies by Turner et al (1954) and De Bresser amp Spiers (1997) whoperformed uniaxial and triaxial compression experiments on calcitesingle crystals Turner et al (1954) performed triaxial deforma-tion experiments from 20 to 400 C confining pressure from 03to 10 GPa and strain rates of 25 times 10minus5 sminus1 in single crystals ofcalcite oriented favorably and unfavorably for twinning In the sam-ples favorably oriented for twinning the yield stresses (sim15 MPa)once extrapolated to higher strain rates using the power-law expres-sion ε prop σ n where n ranges from 93 to 717 (De Bresser Spiers1997) are still an order of magnitude lower than the yield stressesof the 3 μm diameter micropillars favorably oriented for twinning(sim150 MPa) and close to two orders of magnitude lower than theyield stresses calculated from indentation experiments (sim1 GPa)at room temperature The unfavorably oriented samples tested byTurner exhibit yield stresses of sim350 MPa at room temperaturewhich is about a factor of three smaller than the results from boththe micropillar compression experiments and the indentation ex-periments (sim1 GPa) The anisotropy observed in Turner et alrsquoswork is greater than the anisotropy observed in our nanoindentationexperiments We attribute this to the complex stress state duringthe indentation test which requires multiple active slip systems toactivate in order for deformation to occur In contrast a single slipor twin system can accommodate the majority of the deformationin uniaxial and triaxial deformation experiments on single crystals
De Bresser amp Spiers (1997) conducted uniaxial compressionexperiments on calcite single crystals at temperatures ranging from300 to 800 C and strain rates of 3 times 10minus4ndash3 times 10minus8 sminus1 Yieldstresses measured in their study are a factor of 5 less than theyield stresses determined in this study even after correcting forthe difference in strain rate using the same power-law relationshipas above The scale of the samples in the studies by Turner andDe Bresser and Spiersmdashapproximately 12 mm in diametermdashmaybe one reason for the inconsistencies between the results from ourstudy and theirs Further investigation is needed to understand therole of sample size
5 C O N C LU S I O N S
This study used methods from materials science including nanoin-dentation and the first known application of micropillar compressiontesting in geologic materials to constrain the rheology of calcite atlow temperatures The yield stress of calcite depends on both tem-perature and orientation however the temperature dependence isthe dominant effect The data collected in this study agree gener-ally with the data and flow law from previous work (Renner et al2002) These data also provide flow laws that are more confidentlyextrapolated to low temperatures at which natural deformation hasbeen observed The athermal Peierls stress for calcite is determinedby these flow laws to be 12ndash99 GPa depending on which modelis used to convert indentation hardness to uniaxial yield stress andwhich values of p and q are selected The flow laws obtained here
can be used to constrain the rheology of calcite at temperaturesfrom 23 to 175 C thus improving the interpretation of data fromnaturally deformed calcite rocks
A C K N OW L E D G E M E N T S
This work is funded by NSF EAR 1726165 Instrument supportis provided by the Institute of Materials Science and Engineeringat Washington University in St Louis STEM sample preparationwas conducted at the Center for Nanophase Materials Sciencesat Oak Ridge National Laboratory (ORNL) which is a Depart-ment of Energy (DOE) Office of Science User Facility througha user project (AST and RM) Microscopy work performed atORNL was supported by the US DOE Office of Science Ba-sic Energy Sciences Materials Science and Engineering Division(BES-MSED) We would like to thank Andreas Kronenberg and twoanonymous reviewers for their helpful and insightful comments
R E F E R E N C E SBarber DJ amp Wenk HR 1979 Deformation twinning in calcite dolomite
and other rhombohedral carbonates Phys Chem Miner 5 141ndash165Barnhoorn A Bystricky M Burlini L amp Kunze K 2004 The role of
recrystallisation on the deformation behaviour of calcite rocks large straintorsion experiments on Carrara marble J Struct Geol 26 885ndash903
Bharathula A Lee SW Wright WJ amp Flores KM 2010 Compressiontesting of metallic glass at small length scales effects on deformationmode and stability Acta Mater 58 5789ndash5796
Bower AF Fleck NA Needleman A amp Ogbonna N 1993 Indentationof a power law creeping solid R Soc 441 A 97ndash124
De Bresser JHP 1996 Steady state dislocation densities in experimentallydeformed calcite materials single crystal versus polycrystals J geophysRes 101 22 189ndash22 201
De Bresser JHP Evans B amp Renner J 2002 On estimating the strengthof calcite rocks under natural conditions Geol Soc Lond Spec Publ200 309ndash329
De Bresser JHP amp Spiers CJ 1990 High-temperature deformation ofcalcite single crystals by r+ and f+ slip Geol Soc Lond Spec Publ54 285ndash298
De Bresser JHP amp Spiers CJ 1997 Strength characteristics of the r fand c slip systems in calcite Tectonophysics 272 1ndash23
De Bresser JHP Spiers CJ De Bresser JHP amp Spiers CJ 1993 Slipsystems in calcite single crystals deformed at 300ndash800 C J geophysRes 98 6397ndash6409
Burkhard M 2000 Calcite twins their geometry appearance and signifi-cance as stress-strain markers and indicators of tectonic regime a reviewJ Struct Geol 15 351ndash368
Chessin H Hamilton WC amp Post B 1965 Position and thermal param-eters of oxygen atoms in calcite Acta Crystallogr 18 689ndash693
Evans B amp Goetze C 1979 The temperature variation of hardness ofolivine and its implication for polycrystalline yield stress J geophysRes 84 5505
Ferrill DA Morris AP Evans MA Burkhard M Groshong RH ampOnasch CM 2004 Calcite twin morphology a low-temperature defor-mation geothermometer J Struct Geol 26 1521ndash1529
Fischer-Cripps AC 2011 Nanoindentation eds Winer WO et al 3rdedn Springer
Fredrich JT Evans B amp Wong T-F 1989 Brittle to plastic transition inCarrara Marble 94 4129ndash4145
Fredrich JT Evans B amp Wong T-F 1990 Effect of grain size on brittleand semibrittle strength implications for micromechanical modeling offailure in compression J geophys Res 95 10 907ndash10 920
Frost HJ amp Ashby F 1982 Deformation-Mechanism Maps The Plasticityand Creep of Metals and Ceramics Pergamon Press
Ginder RS Nix WD amp Pharr GM 2018 A simple model for indentationcreep J Mech Phys Solids 112 552ndash562 Elsevier Ltd
Dow
nloaded from httpsacadem
icoupcomgjiarticle-abstract22111295691318 by W
ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
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ashington University Law
School Library user on 21 January 2020
Low-temperature rheology of calcite 141
Goldsby DL Rar A Pharr GM amp Tullis TE 2004 Nanoinden-tation creep of quartz with implications for rate- and state-variablefriction laws relavent to earthquake mechanics J Mater Res 19357ndash365
Griggs DT Turner FJ amp Heard HC 1960 Chapter 4 deformation ofrocks at 500 to 800 C Geol Soc Am Mem 79 39ndash104
Hirth G amp Kohlstedt DL 2015 The stress dependence of olivine creeprate Implications for extrapolation of lab data and interpretation of re-crystallized grain size Earth planet Sci Lett 418 20ndash26
Johnson KL 1970 The correlation of indentation experiments J MechPhys Solids 18 115ndash126
Kearney C Zhao Z Bruet BJF Radovitzky R Boyce MC amp OrtizC 2006 Nanoscale anisotropic plastic deformation in single crystalaragonite Phys Rev Lett 255505 1ndash4
Kennedy LA amp White JC 2001 Low-temperature recrystallization incalcite mechanisms and consequences Geology 29 1027ndash1030
Kim S Ree J Han R Kim N amp Jung H 2018 Tectonophysics Fabrictransition with dislocation creep of a carbonate fault zone in the brittleregime Tectonophysics 723 107ndash116
Kocks UF Argon AS amp Ashby MF 1975 Thermodynamics and Kinet-ics of Slip Pergamon Press
Korte-Kerzel S 2017 Microcompression of brittle and anisotropic crystalsrecent advances and current challenges in studying plasticity in hardmaterials MRS Commun 7 1ndash12
Korte S amp Clegg WJ 2009 Micropillar compression of ceramics atelevated temperatures Scr Mater 60 807ndash810
Kranjc K Rouse Z Flores KM amp Skemer P 2016 Low-temperatureplastic rheology of olivine determined by nanoindentation Geophys ResLett 42 176ndash184
Kumamoto KM et al 2017 Size effects resolve discrepancies in 40 yearsof work on low-temperature plasticity in olivine Sci Adv 3 1ndash7
Liu J Walter JM amp Weber K 2002 Fluid-enhanced low-temperatureplasticity of calcite marble Microstructures and mechanisms Geology787ndash790
Mei S Suzuki AM Kohlstedt DL Dixon NA amp Durham WB2010 Experimental constraints on the strength of the lithospheric mantleJ geophys Res 115 1ndash9
Negrini M Smith SAF Scott JM Tarling MS amp Irene M 2018 Mi-crostructural and rheological evolution of calcite mylonites during shearzone thinning constraints from the Mount Irene shear zone FiordlandNew Zealand J Struct Geol 106 86ndash102
Oliver WC amp Pharr GM 1992 An improved technique for determin-ing hardness and elastic modulus using load and displacement sensingindentation experiments J Mater Res 7 1564ndash1583
Oliver WC amp Pharr GM 2004 Measurement of hardness and elasticmodulus by instrumented indentation Advances in understanding andrefinements to methodology J Mater Res 19 3ndash20
Pieri M Burlini L Kunze K Stretton I amp Olgaard DL 2001 Rheo-logical and microstructural evolution of Carrara marble with high shearstrain results from high temperature torsion experiments J Struct Geol23 1393ndash1413
Pieri M Kunze K Burlini L Stretton I Olgaard DL Purg J-P ampWenk H-R 2001 Textrue development of cacite by deformation anddynamic recrystallization at 1000 K during torsion experiments of marbleto large strains Tectonophysics 330 119ndash140
Platt JP amp De Bresser JHP 2017 Stress dependence of microstructuresin experimentally deformed calcite J Struct Geol 105 80ndash87
Renner J amp Evans B 2002 Do calcite rocks obey the power-law creepequation Geol Soc Lond Spec Publ 200 293ndash307
Renner J Evans B amp Siddiqi G 2002 Dislocation creep of calcite Jgeophys Res 107 ECV 6ndash1-ECV 6-16
Rowe KJ amp Rutter EH 1990 Palaeostress estimation using calcite twin-ning experimental calibration and application to nature J Struct Geol12 1ndash17
Rutter EH 1995 Experimental study of the influence of stress temper-ature and strain on the dynamic recrystallization of Carrara marble Jgeophys Res 100 651ndash663
Rybacki E Janssen C Wirth R Chen K Wenk HR Stromeyer Damp Dresen G 2011 Low-temperature deformation in calcite veins ofSAFOD core samples (San Andreas Fault - Microstructural analysis andimplications for fault rheology Tectonophysics 509 107ndash119
Schmid SM Panozzot R amp Bauer S 1987 Special Research Paper lowastSimple shear experiments on calcite rocks rheology and microfabric JStruct Geol 9 747ndash778
Schuster R Scha E Schell N Kunz M amp Abart R 2017 Microstruc-ture of calcite deformed by high-pressure torsion an X-ray line pro fi lestudy Tectonophysics 721 448ndash461
Shaw MC amp DeSalvo GJ 2012 The role of elasticity in hardness testingMetallogr Microstruct Anal 1 310ndash317
Swain M V amp Hagan JT 1976 Indentation plasticity and the ensuingfracture of glass J Phys D Appl Phys 9 2201ndash2214
Tabor D 1970 The hardness of solids Rev Phys Technol 1 145 Retrievedfrom httpstacksioporg0034-66831i=3a=I01
Thom C amp Goldsby D 2019 Nanoindentation studies of plasticity anddislocation creep in halite Geosciences 9 79
Turner FJ Griggs DT amp Heard H 1954 Experimental deformation ofcalcite crystals Bull Geol Soc Am 65 883ndash934
Uchic MD Shade PA amp Dimiduk DM 2009 Plasticity of micrometer-scale single crystals in compression Annu Rev Mater Res 39361ndash386
VanLandingham MR 2003 Review of instrumented indentation J ResNatl Inst Stand Technol 108 249
Verberne BA De Bresser JHP Niemeijer AR Spiers CJ Win-ter DAM amp Plumper O 2013 Nanocrystalline slip zones in calcitefault gouge show intense crystallographic preferred orientation crys-tal plasticity at sub-seismic slip rates at 18ndash150 C Geology 41(8)863ndash866
Vernon RH 1981 Optical microstructure of partly recrystallized calcitein some naturally deformed marbles Tecto 78 601ndash612
Wells RK Newman J amp Wojtal S 2014 Microstructures and rheologyof a calcite-shale thrust fault J Struct Geol 65 69ndash81
Xu L amp Evans B 2010 Strain heterogeneity in deformed Carrara mar-ble using a microscale strain mapping technique J geophys Res 115B04202 doi1010292009JB006458
Mata M Anglada M amp Alcala J 2002 Contact deformation regimesaround sharp indentations and the concept of the characteristic strainJournal of Materials Research 17 964ndash976
Mata M amp Alcala J 2003 Mechanical property evaluation through sharpindentations in elastoplastic and fully plastic contact regimes Journal ofMaterials Research 18 1705ndash1709
