Relation between Effective Conductivity and Susceptibility of Two -- Component Rhombic Checkerboard

TL;DR

This paper establishes a mathematical relation between the effective electric susceptibility and conductivity tensors in a two-component rhombic checkerboard composite, considering different contrast limits and field singularities.

Contribution

It introduces a new relation between effective susceptibility and conductivity tensors for rhombic checkerboard composites, accounting for electric field singularities at corners.

Findings

Derived the relation between effective susceptibility and conductivity tensors.

Analyzed the effects of weak and strong conductivity contrast.

Applicable to thin films and cylindrical samples.

Abstract

The heterogeneity of composite leads to the extra charge concentration at the boundaries of different phases that results essentially nonzero effective electric susceptability. The relation between tensors of effective electric susceptability $\hat\chi_{ef}$ and effective conductivity $\hat\sigma_{ef}$ of the infinite two--dimensional two--component regular composite with rhombic cells structure has been established. The degrees of electric field singularity at corner points of cells are found by constructing the integral equation for the effective conductivity problem. The limits of weak and strong contrast of partial conductivities $\sigma_1,\sigma_2$ are considered. The results are valid for thin films and cylindrical samples.