The Saint-Venant torsion of a circular bar consisting of a composite cylinder assemblage with cylindrically orthotropic constituents

We consider the Saint-Venant torsion of a cylindrical rod of a circular cross section which is filled up by an assemblage of composite circular cylinders. The constituent cylinders consist of a core and a coating both of which are cylindrically orthotropic with the volume fraction of the core being the same in every composite cylinder. The described microstructure is the composite cylinder assemblage of Hashin and Rosen [J. Appl. Mech. 29 (1964) 143] which is now subjected to torsion. The main results are (a) the warping function on the lateral surface of the host rod is zero, (b) an exact expression for the torsional rigidity of the host rod is derived which depends on the size distribution of the composite cylinders but not on their position and (c) there are two circumstances in which the torsional rigidity becomes size distribution independent: The first one is that in which the sizes of the composite cylinders are much smaller than the size of the host rod; the second one is that in which a certain specific relation holds between the properties of the composite cylinder and the volume fraction of the core. If the coating disappears and the core is cylindrically orthotropic, we get the configuration of a polycrystalline rod. Simple bounds for the torsional rigidity of the constructed composite rod are obtained.