Using differential geometry to characterize and analyse the morphology of joints

Abstract

Abstract Concepts of differential geometry are reviewed, and it is demonstrated through examples that the main joint surface, rib marks and hackle of a joint may be described using parametric representations such that the first and second fundamental forms fully characterize these surfaces. Other useful quantities are the unit normal vector, the principal normal curvatures, and the Gaussian and the mean curvature. Sufficiently close to any point on a surface the shape is planar, parabolic, elliptical or hyperbolic. The surface of a joint in chert and another in siltstone were scanned and the resulting data analysed. Although the main joint surface of the chert sample is approximately planar, it is composed of low-amplitude undulations with elliptical and hyperbolic forms. The unit normal vector does not vary by more than about 3.4° over this surface, which is consistent with the threshold angle for the initiation of hackle based on laboratory experiments. An individual hackle is found to be approximately helicoidal in shape, but only in the breakdown zone. Rib marks on the siltstone sample have distinct and similar morphologies, with a concave base and convex peak. Field and laboratory campaigns designed to test hypothese about the geometry of joints should use the principles and tools of differential geometry.