Effective Conductivity of Spiral and other Radial Symmetric Assemblages

TL;DR

This paper investigates the effective electrical conductivity of various complex, radially symmetric micro-structures, including spirals and assemblies with hierarchical laminates, revealing unique differential equations and field transformations.

Contribution

It introduces a coupled differential equation model for spiral assemblages and compares their effective conductivity to classical geometries, extending homogenization theory.

Findings

Fields in spiral assemblages satisfy coupled second order differential equations.

External homogeneous fields are rotated inside the inclusions.

Effective conductivity of 2D Star assemblies matches Hashin-Shtrikman coated circles, but 3D Spiky Balls differ from coated spheres.

Abstract

Assemblies of circular inclusions with spiraling laminate structure inside them are studied, such as spirals with inner inclusions, spirals with shells, assemblies of "wheels" - structures from laminates with radially dependent volume fractions, complex axisymmetric three-dimensional micro-geometries called Connected Hubs and Spiky Balls. The described assemblages model structures met in rock mechanics, biology, etc. The classical effective medium theory coupled with hierarchical homogenization is used. It is found that fields in spiral assemblages satisfy a coupled system of two second order differential equations, rather than a single differential equation; a homogeneous external field applied to the assembly is transformed into a rotated homogeneous field inside of the inclusions. The effective conductivity of the two-dimensional Star assembly is equivalent to that of Hashin-Shtrikman coated circles, but the conductivity of analogous three-dimensional Spiky Ball is different from the conductivity of coated sphere geometry.