A Green’s function approach is used to formulate and obtain the stress field, under torsional loads in a radially finite solid cylinder with radially variable elastic modulus. With this approach a certain dual static-geometric analogy in the solution is readily proved and applied to generate the solution with stress boundary conditions from that with displacement boundary conditions and vice-versa. The problem is solved using both boundary conditions and for an exponentially varying shear modulus. In particular, under displacement boundary conditions, the stress field in the solid with a generalised Reissner-Sagoci boundary condition is easily deduced. With stress boundary conditions, the criteria for crack propagation in such elastic models are also obtained using the Griffith-Irwin condition of rupture.