Modeling polycrystals with regular polyhedra

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.

Estimation and Utilization of Structure Anisotropy in Forming Pieces

“…Therefore deformation of various idealized grain shapes can be investigated. For instance crystals can be modeled by regular polyhedron -tetrakaidecahedron [2]. One method is based on dependence of the ratio of relative surface area of grain boundaries in deformed state S V and undeformed state S V0 to strain [3].…”

“…In case of anisotropic plastically deformed structure the grains have anisometric dimension, it is necessary to describe their anisotropy [1]. The anisotropic microstructure is decomposed into isotropic, planar and/or linear oriented components -specific surface area of grain boundaries and these parameters are measured using stereology [2]. Degree of grain boundary orientation is estimated as ratio of oriented specific surface area to total specific surface area.…”

Topological and metrical aspects of normal and abnormal grain growth

“…The construction of ANHs has been described in detail in earlier publications [2][3][4][5][6][7] . Two ANHs, one with four faces, the 4-hedron, and another with six faces, the 6-hedron, are illustrated in Figure 1.…”

“…Expressions for topological factors corresponding to the (dimensionless) curvature, h, area, a, and volume, v, are not given here for space reasons. But full details of the structure and mathematical properties of ANHs may be found elsewhere [2][3][4][5] .…”

“…This approach greatly simplifies the analysis of materials configured as real irregular networks, such as polycrystals, as all the geometric properties of ANHs can be calculated exactly. ANHs may, therefore, be used as 'proxies' for analyzing real irregular network grains, allowing the rigorous treatment of related problems pertaining to such network structures [2][3][4][5][6][7] .…”