EBSD-Based Calibration of Differential Stress From Experimentally Deformed Black Hills Quartzite Using the Perimeter-Area Fractal Dimension

Microstructures that preserve quantitative information about deformation conditions during viscous flow of crystalline materials are rare, the most common example being the dynamically recrystallized grain or subgrain size to infer differential stress. There are many instances in which identification of recrystallized grains or subgrains is challenging. Thus, another microstructural attribute that relates to differential stress would be useful. We use electron backscatter diffraction (EBSD) data from experimentally deformed Black Hills quartzite to show that the perimeter-area fractal dimension of quartz aggregates (the slope of the log-log relationship between perimeter and diameter), which we term the grain boundary dimension (GBD), strongly correlates with differential stress (σ) in rocks deformed by dislocation creep. Unlike traditional methods for estimating differential stress, the GBD method does not require identification of recrystallized grains or subgrains. Analysis of 9 samples yields both power-law and logarithmic calibrations (σ in MPa) $GBD=\left(0.691\pm 0.098\right)\times {\sigma }^{\left(0.105\pm 0.031\right)}$ and $GBD=\left(0.267\pm 0.081\right)\times \log \left(\sigma \right)+\left(0.59\pm 0.16\right)$, which show excellent agreement with published grain-size piezometers. Analysis of kernel average misorientation maps, guided by theoretical considerations, suggests that GBD develops through heterogeneous grain boundary migration related to spatial variation of driving force from dislocations and dislocation walls in adjacent grains. Our calibrations cover dislocation-creep conditions in which local grain boundary migration and subgrain rotation are the main dynamic recrystallization processes. Further work is needed to refine the calibration, test extrapolation to natural conditions, assess its applicability to general-shear deformation, and extend the method to other minerals such as calcite, rock salt, olivine and ice.