Strong influence of a small fiber on shear stress in fiber-reinforced composites

In stiff fiber-reinforced composites, it has been known that the shear stress increases at the rate of 1/ϵ as the distance ϵ between adjacent fibers approaches 0. This paper reveals a strong influence of a combination of a triple fiber, as well as the distance between a pair of fibers, on the blow-up so that the stress concentration can be significantly accelerated by adding a small fiber in-between fibers. Specifically, if a fiber F2 with a small diameter δ is located in-between fibers F1 and F3, ϵ1=dist(F1,F2) and ϵ2=dist(F2,F3), then the stress blows up at the exact rates of 1/δϵ1 and 1/δϵ2 between F1 and F2 and between F2 and F3, respectively. This estimate still holds even when a part of F2 overlaps with F3. The magnification factor 1/δ yields the enormous increase in the stress that greatly surpasses the expectancy by previous methods.