  In stiff fiber-reinforced material, the high shear stress concentration
occurs in the narrow region between fibers. With the addition of a small
geometric change in cross-section, such as a thin fiber or a overhanging part
of fiber, the concentration is significantly increased. This paper presents
mathematical analysis to explain the rapidly increased growth of the stress by
a small particle in cross-section. To do so, we consider two crucial cases
where a thin fiber exists between a pair of fibers, and where one of two fibers
has a protruding small lump in cross-section. For each case, the optimal lower
and upper bounds on the stress associated with the geometrical factors of
fibers is established to explain the strongly increased growth of the stress by
a small particle.
