TWO TYPES OF ELECTRIC FIELD ENHANCEMENTS BY INFINITELY MANY CIRCULAR CONDUCTORS ARRANGED CLOSELY IN TWO PARALLEL LINES

In stiff fiber-reinforced composites, high shear stress concentrations occur in narrow regions between neighboring fibers. Electric fields and conductors correspond to shear stresses and cross-sections of fibers respectively in the anti-plane shear model. Due to material failure of composites, there have been intensive studies so far to estimate an electric field in between only a finite number of conductors. Indeed, the composites contain a large number of stiff fibers, and the concentration can be strongly enhanced by some combinations of inclusions. Two types of enhancements in parallel and perpendicular directions by a combination of infinitely many inclusions in a line are the main subjects of this paper. We thus consider the electric fields in between a infinite number of perfectly conducting unit disks arranged closely and regularly in two parallel lines. Asymptotes and optimal blow-up rates for the fields in two kinds of narrow regions are obtained in terms of the distances between conductors. In particular, very strong enhancement in the parallel direction is exhibited to have the blow-up rate substantially different from the existing result in the case of finite inclusions.