Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections

We consider high stresses in stiff-fiber reinforced materials, which increase rapidly as fibers approximate to one another. This paper presents the optimal blow-up rate of the stresses with respect to the distance between a pair of stiff fibers in R3. The blow-up result plays an important role in our understanding of low strengths of fiber-reinforced composites. Referring to a problem of anti-plane shear, the stresses can be interpreted as the electric fields outside closely spaced perfect conductors in R2, under the action of applied electric field ∇H. It has been shown by Ammari, Kang et al. that in the particular case of circular inclusions, the electric field blows up at the optimal rate ϵ−1/2 as ϵ→0, where ϵ is the distance between conductors. Recently, Yun has extended the blow-up result to pairs of conductors associated with a large class of shapes whose complements can be transformed conformally to the outside of a circle with C2 mapping. However, it presented a suboptimal result that only for a special uniform field ∇H=(1,0), the electric fields blow up at the exact rate ϵ−1/2. In this paper, an upper bound with the rate ϵ−1/2 of electric field for any harmonic function H is established. This yields the optimal blow-up rate ϵ−1/2 for the inclusions in the same class of shapes as Yun.