Exact solutions in torsion of composite bars: Thickly coated neutral inhomogeneities and composite cylinder assemblages

The study of microgeometries amenable to exact solutions has been of considerable interest in the modern literature of composite media. Although several such solutions exist today in the context of applied uniform loadings, the existence of exact solutions in more complex settings like torsion of cylindrical bars has been virtually unexplored. In the present paper we study the Saint-Venant torsion of composite cylindrical bars and derive exact solutions in two stages. In the first stage, we show the existence of thickly coated inhomogeneities of various shapes that leave the vanishing warping function in the host circular bar undisturbed. These are called 'partly neutral' inhomogeneities. If the torsional rigidity of the host bar is left unchanged as well, they are called 'completely neutral'. The host bar can accommodate several such inhomogeneities. In the second stage, we make use of partly neutral coated circular inhomogeneities and derive an exact solution to a circular cylindrical bar in torsion that is filled up by the renowned composite cylinder assemblage (CCA) of Hashin & Rosen. The method of analysis in the first stage is based on complex function theory and conformal mapping. If the coated inhomogeneities have a vacuous core, then they are represented by a Laurent series expansion, which maps the coated cavity in the physical plane onto a circular annulus in the complex plane. The topological properties of such coated cavities are established and a rich class of examples are given. Coated inhomogeneities with a non-vanishing shear modulus of the core are represented by a mapping with positive powers only in the series expansion. The derived results assume extremely simple forms if the coated inhomogeneities are elliptical. It is shown that in order to achieve neutrality a coated ellipse needs to be positioned such that its major axis lies on a radial line of the circular rod. The obtained exact solutions for the CCAs with circular cylinders include two-phase and multiphase systems. We show that the torsional rigidity of such a composite bar depends on the size distribution of the composite cylinders that fills completely the circular rod. The torsional rigidity becomes independent of this size distribution if a certain condition is obeyed between the volume fraction of a typical composite cylinder and the shear moduli of the core and coating.