Rotation and strain of linear and planar structures in three-dimensional progressive deformation

Abstract

Rotation and progressive strain have been studied for a sheet embedded in a matrix which undergoes rotational three-dimensional strain under constant volume conditions. The mathematics gives explicit information on the following features: 1. (1) The length and position (relative to a defined coordinate system) of the principal axes of the strain ellipsoid at any stage of the progressive deformation. 2. (2) The position and length of the principal axes in any plane intersecting the strain ellipsoid, also at any stage of the deformation. 3. (3) The position and length of passive markers which initially coincided with the principal axes in an intersecting plane. This is of consequence for the distinction between passively rotating structures and actively forming structures. 4. (4) The shear strain parallel to an intersecting plane or sheet, as indicated by the angular difference between the normal to an intersecting plane at any time and the marker at the same time which initially, however, was parallel to the normal. This layer-parallel shear causes boudins to rotate and the axial plane of buckles to tilt. The relationships have been expressed quantitatively in the bulk of the paper and illustrated in diagrams. The analysis presented is basic for the study of the deformational behavior of competent sheets of rocks embedded in less competent ones.