Statistical method for determining the flat segments of boundaries for crystallites

A method for specifying flat sections of crystallite boundaries (grains or sub-grains), taking into consideration their topology, is proposed. The shape of crystallites in the form of ellipsoids is assumed to be known based on the approximation of averaged experimental data. A statistical problem on the determination of flat sections of grain boundaries, the area, orientation and position in space of which are found taking into account the shape of the prescribed ellipsoid, is formulated. To solve this problem, an attempt to ensure the fulfillment of two conditions is made. The first one is flat sections of the boundary along the surface of the ellipsoid should have an approximately uniform distribution. The second one is that in places of the greatest curvature of the ellipsoid surface there is an increased density of facets with the smallest area. The noted conditions provide the necessary approximation of the ellipsoid by flat sections for the implementation of the developed statistical multilevel models, taking into consideration the structure of the material.