Stereology of plane convex loops distributed in the three-dimensional microstructure

Lineal microstructural features are present in almost all the microstructures. Such features often have a plane convex shape, and they can be regarded as plane convex loops. A stereological estimator is developed for the estimation of the sum of the squares of the areas of plane convex loops per unit volume, from the measurements performed on the plane of polish. It is shown that the average value of the cubes of the chord lengths resulting from the intersections of the plane convex loops with the plane of polish is directly related to the sum of the squares of the areas of plane convex loops per unit volume. The result is also applicable to any ensemble of plane convex loops of different shapes and sizes. The applications of this new stereological attribute for the characterization of microstructure and microstructural evolution are discussed.