Analysis of 3-D network structures

Abstract

Integral geometry and topology are applied to the problem of space-filling in irregular network structures such as liquid and amorphous metals, polycrystals, and foams. The theory developed is based on representing cells in irregular polyhedral networks as “average

-hedra”, where

equals the number of faces between contacting neighbours. Average

-hedra satisfy, at their edges and vertices, the network's angular averages for triple lines and quadrajunctions, as dictated by topology. Although they themselves are incapable of filling space as a contiguous network, average

-hedra act as high-symmetry “proxies” for analysing the average metric, energetic, and kinetic behaviour of real irregular network cells of equivalent topologies. The analysis developed here accurately predicts the average behaviour of 3-dimensional network structures, such as polycrystals and foams, and may also be applicable to biological tissues. This approach should prove especially useful for constructing quantitative descriptions of evolving microstructures. The analytic relations derived here can provide precise benchmarks to test numerical simulations of the properties of constructible (irregular) network cells, and can guide future quantitative experiments and numerical modelling.

Acknowledgement

The author acknowledges the financial support received during 2002–2003 as an Alexander von Humboldt Preissträger from the Alexander von Humboldt Stiftung, Bonn, Germany, permitting his study and scientific interactions at the Institut für Metallkunde und Metallphysik (IMM), Rheinisch-Westfälische Technische Hochschule, Aachen, Germany. The author is indebted to his IMM colleagues Professors Lasar Shvindlerman, Anthony Rollett, and Günter Gottstein, all of whom advocated the need for developing an analytical theory for 3-dimensional grain growth. The author also thanks Dr. Gerda Pomana, IMM-Aachen, Germany; Professors Adam A. Wheeler, Faculty of Mathematical Studies, University of Southampton, UK; Peter Streitenberger, Otto von Güricke University, Magdeburg, Germany; and Robert F. Sekerka, Carnegie-Mellon University, Pittsburgh, PA, USA, for their helpful suggestions offered early in the development of this work. The author is especially grateful for the detailed mathematical discussions held with Professor Nicholas Alikakos, Department of Mathematics, University of Athens, Greece; Professor David Wu, Department of Mechanical Engineering, Yale University, New Haven, CT; and Professor Frans Spaepen, Division of Applied Science, Harvard University, Cambridge, MA, all of whom helped clarify points concerning the statistical and geometrical properties of n-dimensional networks, and for performing some additional analyses that led to better insights into the behaviour of average

-hedra. Finally, the author expresses his gratitude to Professor Afina Lupulescu, Rensselaer Polytechnic Institute, for her able assistance in preparing the manuscripts and illustrations.

Notes

Email: [email protected].

Additional information

Notes on contributors

M. E. Glicksman †

Email: [email protected].