An exactly solvable microgeometry in torsion: Assemblage of multicoated cylinders

An exact expression for the torsional rigidity of a cylindrical bar with arbitrary transverse cross-section filled with an assemblage of multicoated inclusions is derived. The exact formula depends on the constituent shear rigidities, the area fractions and the size distribution of the multicoated inclusions, but is independent of the assembly microstructure. The analysis is based on a successive construction of neutral multicoated inclusions under Saint-Venant's torsion. We show how to design permissible multicoated inclusions, with phase-shear rigidities and area fractions appropriately balanced, so that after its introduction into a homogeneous host bar the warping field in the host bar will not be disturbed. What makes the neutral inclusion under torsion particularly intriguing is that both the constraint conditions and the torsional rigidity are independent of the location of the neutral inclusion. One can thereby add many neutral inclusions to fill up the cross-section without further derivations. Without solving any field equations, we prove that the torsional rigidity of a given cross-section filled with an assemblage of multicoated inclusion can be exactly determined in a simple, explicit form.