Torsion of an isotropic shaft of arbitrary cross-section embedded with multicoated or graded circular cylinders of cylindrically orthotropic materials

Finding a geometric configuration that is amenable to an exact determination or characterization of the torsional rigidity is a relatively new territory that has only recently begun to be explored. For example, a circular cross-section with an assemblage of composite cylinders was only recently known to be an exactly solvable microgeometry. A host shaft with arbitrary crosssection, equivalent to higher orders of boundary data, necessitates that the coated cylinder be sufficiently multiply coated. The present analysis is to provide a theoretical framework showing how to design a neutral cylinder with any number of coatings or with graded shear moduli in a cross-section under torsion. Specifically we consider that the constituents are cylindrically orthotropic with the shear moduli μ_r and μ_θ. The host shaft is isotropic with the shear modulus μ₀. A simple and unified mathematical framework is first proposed for the analysis of a multicoated cylinder. It is proven that only a two by two matrix, resulting from a serial multiplication of matrices of the same order, will enter into the resulting expression. Next, the multicoated cylinder, which consists of piecewise constant shear moduli, is generalized into a graded cylinder, with a continuous variation of the shear modulus along the radial direction. We find that the warping field of a neutral graded cylinder, with varying radial and tangential shear moduli, is governed by a second-order, homogeneous, ordinary differential equation. The method of Frobenius is adopted to obtain series solutions for the warping functions. An interesting result for the neutrality of the embedded cylinders (multicoated or graded) is that when the geometric mean of the shear moduli μ_G = √μ_rμ_θ is identical to the shear modulus μ₀ of the host shaft, the neutrality condition is satisfied for any cross-section of the host shaft. Finally, a condition is given for the torsional rigidity of the host shaft to remain the same with the inclusion of the embedded cylinders.