A kinematic model of preferred orientation
Kinematic models of mineral grain behavior during deformation have been provided by G. B. Jeffery9s study of rigid ellipsoids in a viscous fluid and A. March9s analysis of deformable rods and plates. In this paper, a more general kinematic model is presented which is applicable to arbitrarily shaped grains. The simple premise of a constant linear relationship between the angular velocity of a mineral grain and the macroscopic strain rate of its surrounding material leads to the linear model. A further premise involving certain kinematic properties of the volume of deforming material displaced by an embedded mineral grain leads to a special form of the linear model called the displacement model; this model provides specific equations describing the motion of an individual mineral grain of arbitrary shape. The displacement model coincides with Jeffery9s and March9s models in the special cases of ellipsoids and of ideally slender needles or ideally thin plates, respectively. In addition, the displacement model is in good agreement with experimental measurements of the motions of rigid bodies of various shapes suspended in a slowly flowing viscous fluid. Using the displacement model for a particular class of grain shapes, it is possible to compute the preferred orientation diagrams resulting from an arbitrarily given strain history. Several sequences of such diagrams are presented showing the development of preferred orientation for different shapes and strains. These diagrams have the property that they depend not only upon the grain shape, but also upon the strain history. This is in contrast to March9s solutions, which depend solely upon the finite strain ellipsoid and are otherwise independent of the strain history. The specific properties of the preferred orientations produced by the displacement model can be tested against observations of naturally occurring mineral orientations. In order to be significant, however, such observations will require simultaneous measurements of both grain shapes and grain orientations, a task which has apparently been attempted only by W. A. Duffield. His results are reviewed and are shown to be in agreement with the displacement model in the area studied.