Finite Torsion of Compressible Circular Cylinders: An Approximate Solution

This paper deals with the equilibrium problem of circular cylinders under finite torsion. A three-dimensional kinematic model, where the large twisting of the cylinder is accompanied by transverse contraction and longitudinal extension, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains as unknowns the longitudinal displacement, the rigid rotation and the transverse stretch of cross sections. To simplify the mathematical formulation, the transverse stretch is assumed to be constant, as it radially undergoes very low variations. This hypothesis produces some approximations in the field equations, but the equilibrium solution obtained is however characterized by a satisfactory accuracy, as shown by the comparisons performed using the numerical techniques of the Finite Element Method (FEM). A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the unknowns are determined by imposing the equilibrium conditions and the boundary conditions. For the end base of the cylinder two different boundary conditions have been considered, according to which the longitudinal translation of this surface is allowed or prevented. Once the kinematic unknowns have been determined, explicit formulae for displacements, stretches and stresses are provided, which show the role of the geometric and constitutive parameters, as well as of the twisting angle. The results provided by the proposed solution are shown by a series of graphs. The same torsion problem has been addressed with FEM. A very good agreement was found between the results obtained with the two different analyses. Finally, the nonlinear torsion problem was linearized by introducing the hypothesis of smallness of the displacement and deformation fields. With this linearization, the classical solution for the infinitesimal torsion problem was fully retrieved.