Torsion and flexure of inhomogeneous elements

The torsion of bars with inhomogeneous shear moduli is considered, under different assumptions on the cross-sectional geometry, and the spatial variation law for the moduli. In particular, bars of arbitrary geometry, and modulus that varies in the section only through a function of the coordinates are considered, under the additional assumption that the modulus is constant on the sectional boundary. Bounds on the torsional rigidity are established. As special cases, laminates are discussed, and novel solutions for the torsion of a circular cylindrical bar with angular symmetry are presented. The problem of flexure of inhomogeneous cylindrical bars is considered, allowing for arbitrary variation of the moduli but with constant Poisson's ratio. Specific solutions are exhibited for the case of angular and radial variations of the shear modulus. Finally, the derived solutions are employed to determine the effective properties of the graded structures.