DEFORMABLE AND RIGID ELLIPSOIDS IN VISCOUS FLOWS: COMPUTATIONAL TECHNIQUES

The dynamic theory of rigid and deformable ellipsoidal inclusions in viscous flows was worked out by Jefferey (1922) and Eshelby (1957, 1959), and further developed and applied by various authors. Recently, geologists have become interested in applying this theory to the analysis of paleomagnetic data, shape preferred orientation data, and other field data, in recognition of the fact that viscosity contrasts between inclusions and their host rocks are important to the quantification and interpretation of deformation. The analysis of such field data therefore benefits from improvements in the speed and precision of computing rigid and deformable ellipsoid dynamics.

Using Eshelby's theory, we describe three approaches to computing deformable ellipsoids and two approaches for computing rigid ones. The most sophisticated of our methods use differential-geometric techniques on Lie groups. (In mathematics, a Lie group is a space of transformations, such as the space of all volume-preserving, homogeneous finite deformations.) We achieve roughly a ten-fold reduction in error, or a nine-fold increase in speed, over earlier methods. Also, our methods perfectly preserve certain geometric properties of the ellipsoid, such as the volume. In contrast, other, non-Lie-group methods may unwittingly allow the ellipsoid volume to change, and may even allow the ellipsoid to degenerate to a cylinder or hyperboloid. We apply our ellipsoid techniques to the analysis of shape preferred orientation data from the mantle section of the New Caledonia ophiolite, where orthopyroxene grains in an olivine matrix are deformed in a shear zone.