Orthographic analysis of geological structures—I. Deformation theory

Abstract

Orthographic projection transforms a three-dimensional object into its two-dimensional image by a mechanism of homogeneous deformation, with infinite shortening along the parallel projection lines. The orthographic orientation net is therefore a useful tool for the analysis of rank-2 tensor operations such as deformation and displacement. A tensor transforms orthogonal radii of the unit sphere into conjugate radii of an ellipsoid, here termed the tensor's ellipsoid. In two dimensions, any symmetric or asymmetric tensor's ellipse may be represented by a great circle on the orthographic orientation net or by a Mohr circle, generally in an ‘off-axis’ position. The Mohr circle's points of intersection with its reference axis define the tensor's eigenvectors and eigenvalues, which may be real or complex. An asymmetric tensor's eigenvectors are not mutually orthogonal, nor do they parallel the principal semiaxes of the tensor's ellipse. These facts are used to classify plane deformations. Any rotational deformation may be factored using Mohr circles for polar or additive decomposition. In three dimensions, a symmetric or asymmetric tensor's ellipsoid may be represented by a graticule on the orthonet with the aid of polar decomposition. The two-dimensional tensor and ellipse corresponding to any section of the three-dimensional state may be determined from a skiodrome. In order to apply rank-2 tensor concepts to heterogeneous deformation, a way of describing the gradients of rank-2 tensors (i.e. their variation with position in the heterogeneous tensor field) is needed. A rank-3 tensor serves this purpose.