Analysis of 3-D network structures

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.