For compressible materials, the mechanical response to a torsional deformation (at sufficiently large levels of deformation) includes the nonlinear effects of a compressive normal force along the axis of the cylinder and a radial expansion of the cylinder, in addition to the expected torque response. The National Institute of Standards and Technology torsional dilatometer [Duran and McKenna, (1990)] has been used to measure simultaneously the torque, the axial normal force, and the volume change in response to a torsional deformation. In stress-relaxation experiments with an epoxy cylinder just below its glass transition temperature, the torque and normal force decay monotonically, but the volume change associated with the torsion shows an extended relaxation behavior with significant nonmonotonic decay at the lowest temperature investigated. The measurements are modeled with a series solution for torsion of an elastic, compressible material [Murnaghan (1951)]. The elastic solution is adapted for viscoelastic behavior by assuming that isochronal data can be treated as equilibrium elastic data, following a suggestion of Rivlin (1956).