Generalized Hough transform for the stress inversion of calcite twin data

Since the mechanical twinning along calcite e-planes has a critical resolved shear stress, not only principal stress axes but also differential stress can be determined from the orientations of twin lamellae. Based on the five-dimensional stress space that fulfills the principle of coordinate invariance, it is shown in this article that the inversion of twin and untwin data is comparable with fitting a spherical cap to data points on a unit sphere in the space. The principal stress orientations and stress ratio are indicated by the center of the cap, whereas differential stress is denoted by the size of the cap. Based on this geometrical interpretation, the generalized Hough transform was applied to the inversion of the data in this study. The new method is demonstrated to be robust to sampling bias, variability in the critical resolved shear stress. The determination of differential stress was difficult when the differential stress to be detected was ∼10 times larger than the critical resolved shear stress. Stresses were separated by the method from heterogeneous data successfully as long as the spherical caps corresponding to the stresses to be detected had no or a small intersection.