ON FOLDING DURING THREE-DIMENSIONAL PROGRESSIVE DEFORMATION

This paper is concerned with the geometry of homogeneous strain in rocks. Equations relating rotation of planes and lines to the deformation of the rock containing them are derived with the aid of the deformation ellipsoid. The Fresnel construction is borrowed from optical mineralogy to determine from the deformation ellipsoid the principal axes of compression and extension in any plane, and an expression is derived for determining the lengths of these axes. Further equations are derived which define the surfaces of no infinitesimal strain and no finite strain for all possible ellipsoids, and it is shown that these surfaces may be used for rapidly determining whether the principal axes in any plane have suffered shortening or extension. With these tools the rather complex geometry of three-dimensional homogeneous strain is examined in detail. It is shown, for instance, that during deformation planes and lines develop preferred orientations reflecting the symmetry of the deformation, and pre-existing folds rotate bodily in space and either open or close or both during the deformation. In order to draw further conclusions of general interest the condition of homogeneous strain is relaxed in order to consider the deformation of layered rocks with competence differences between the layers. It is argued that during the deformation of such rocks folds and boudinage form parallel to the principal directions of strain in the layers. The conditions that determine whether folds or boudinage are formed are examined and the subsequent development of the structures after their generation is followed. These methods of analysis are used in the construction of models of superimposed fold-systems which are shown to be similar to some recently described field examples. It is argued that the tectonic axial cross must be replaced by the deformation ellipsoid. The structures treated in the paper are usually derived by shear-plane hypotheses, but it is considered that such hypotheses are superfluous.