Bounds and extremal configurations for the torsional rigidity of coated fiber reinforced shafts

In this paper we derive bounds on the torsional rigidity for coated fiber reinforced shafts. The bounds are used to assess the optimality or suboptimality of fiber reinforcement configurations. This investigation focuses on coated fiber reinforcements with circular cross section. It is shown how the effective antiplane shear modulus and torsional rigidity of each coated fiber are used to determine whether the configuration provides reinforcement above or below that of a homogeneous shaft containing no coated fibers. Simply connected shaft cross sections of arbitrary shape reinforced with any configuration of coated fibers are considered. Precise conditions on the effective antiplane shear modulus and torsional rigidity of each coated fiber are given under which the circular shaft reinforced with a single centered circular coated fiber is either optimal or suboptimal.