Abstract
Polycrystalline structure is of paramount importance to materials science and engineering. It provides an important example of a space-filling irregular network structure that also occurs in foams as well as in certain biological tissues. Therefore, seeking an accurate description of the characteristics of polycrystals is of fundamental importance. Recently, one of the authors (MEG) published a paper in which a method was devised of representation of irregular networks by regular polyhedra with curved faces. In Glicksman's method a whole class of irregular polyhedra with a given number of faces, N, is represented by a single symmetrical polyhedron with N curved faces. This paper briefly describes the topological and metric properties of these special polyhedra. They are then applied to two important problems of irregular networks: the dimensionless energy 'cost' of irregular networks, and the derivation of a 3D analogue of the von Neumann-Mullins equation for the growth rate of grains in a polycrystal.
Highlights
The polycrystalline structure is of paramount importance to materials science and engineering
It is an example of a space-filling irregular network structure that occurs in foams as well as in certain biological tissues
The technological importance of polycrystals derives from the fact that the majority of industrial crystalline materials are used as polycrystals
Summary
The polycrystalline structure is of paramount importance to materials science and engineering. A complete description of such a network involves the knowledge of the geometric characteristics of the individual crystals, of their crystallographic orientation, i.e., their crystallographic texture, of the nature of the interfaces between individual crystals, among others. Is it important to know these characteristics at one instance in time, and to predict their dynamic behavior when such structures change as a function of time because of grain growth. In Glicksman’s method a whole class of irregular polyhedra with a given number of faces, N, is represented by a single symmetrical polyhedron with N faces These polyhedra are ‘regular polyhedra’ with curved faces, constructed in such a way that they satisfy the average topological constraints. This paper will focus on the properties of ANHs rather than on the detailed mathematical derivations that can be found elsewhere[1,2,3,4,5]
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