USING LASER SCANNING AND DIFFERENTIAL GEOMETRY TO CHARACTERIZE AND ANALYZE THE MORPHOLOGY OF JOINTS

Joint surfaces record the propagation history of a mode I fracturing event in the complex and often beautiful morphologies seen upon them. We have combined digital scanning with differential geometry to quantitatively explore the surfaces of two joints with a focus on rib marks and hackle. The laser scanning provided a high-resolution (<0.200 mm accuracy) data set of position vectors for each joint surface. Differential geometry provided the mathematics to quantitatively describe the local shape and orientation of the surface near each data point. A filter based on 2-D Fourier analysis was used to selectively remove designated ranges of wavelengths from the data after re-sampling the surface to a grid using linear interpolation. Two filtering strategies were employed: 1) minimal, to remove only surface “noise” resulting from debris and damage; and 2) maximal, to remove all but the low frequency structures. Residuals of the maximal filtering accentuate minor joint surface features, such as incipient hackle. Curvature of the minimally filtered surfaces was calculated at each grid point using the first and second fundamental forms of differential geometry, supplying the Gaussian and mean curvatures, the principle curvature directions and values, the normal to the surface, and the shape curvature (elliptical, hyperbolic, parabolic or planar). Parabolic and planar shapes are defined by one or both principal curvatures being identically zero, an unlikely occurrence in sampled data, so a method was devised to determine a threshold under which curvatures are assigned a statistically “zero” value. Main joint surfaces were found to be predominantly planar, with local parabolic, hyperbolic and elliptical shapes. Hackle and cross fractures on both sample surfaces have similar shapes despite significant differences in size, with the hackle having smooth planar faces. Rib marks have been described qualitatively in the literature as both cusped and asymmetrical with a convex face in the direction of joint propagation and a concave face nearest the main joint surface. Unlike these shapes, the rib marks examined here have distinct and similar morphologies, with concave bases and convex peaks. This quantitative description of joint surfaces lends insight into the fracturing process.