Two types of electric field enhancements by infinitely many circular conductors arranged closely in two parallel line

TL;DR

This paper investigates the electric field concentration between infinitely many circular conductors arranged in two parallel rows, revealing two distinct types of field enhancement and establishing asymptotic behaviors and blow-up rates.

Contribution

It introduces new asymptotic estimates and blow-up rates for electric fields in configurations with infinitely many conductors, extending finite inclusion results to an infinite setting.

Findings

Identifies two types of electric field enhancement effects.

Establishes asymptotic formulas for the field in narrow regions.

Finds a blow-up rate significantly different from finite inclusion cases.

Abstract

In this paper, we consider very high concentration of electric field in between infinitely many circular perfect conductors arranged closely in two rows. In stiff fiber-reinforced composite, shear stress concentrations occur in between neighboring fibers, and the electric field means shear stress in this paper. Due to material failure of composites, there have been intensive studies so far to estimate the field in between only a finite number of inclusions. Indeed, fiber reinforced composites contain a large number of stiff fibers, and the concentration can be strongly enhanced by some combination of inclusions. Thus, we establish some asymptotes and optimal blow-up rates for the field in narrow regions in between infinitely many conductors in two rows to describe the horizontally and vertically combined effects of a large number of ones. Especially, one of the blow-up rates is substantially different from the existing result in the case of finite inclusions.