This work is concerned with deriving the upper and lower bounds for the torsional rigidity of cylindrical shafts with arbitrary cross-section containing a number of multiply coated fibers with imperfect interfaces along the interfaces. Each multicoated fiber may have different constituents with different area fractions. In the formulation, we first extend our previous formulation, based on classical energy principles in elasticity, to construct torsional rigidity bounds for shafts containing simply coated fibers with two different kinds of imperfect interface. Next, based on the present results for shafts containing simply coated fibers and our previous findings for shafts containing homogeneous fibers with imperfectly bonded interfaces, we propose a concept of replacement fiber with an effective shear rigidity to replace the effect of fiber with imperfect bonding interface. In addition, we propose an equivalent shear rigidity to simulate the effect of a simply coated fiber. This replacement procedure allows us to construct the bounds, through a recursive procedure, for the torsional rigidity of shafts containing multiply coated fibers with possibly imperfect interfaces.
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