Large twisting of non-circular cylinders in unconstrained elasticity

This paper deals with the equilibrium problem of non-circular cylinders subjected to finite torsion. A three-dimensional kinematic model is formulated, where, in addition to the rigid rotation of the cross sections, the large twist of the cylinder also generates in- and out-of-plane pure deformation of the cross sections and the variation of the cylinder length. Following the semi-inverse approach, the displacement field prescribed by the above kinematic model contains an unknown constant, which governs the elongation of the cylinder, and three unknown functions which describe the pure deformation of the cross sections. A Lagrangian analysis is then performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the boundary value problem is formulated. Nevertheless, the governing equations assume a coupled and nonlinear form which does not allow to apply standard solution methods. Therefore, the unknown functions are expanded into power series using polynomial terms in two variables. These series contain unknown constants which are evaluated applying the iterative Newton's method. With this procedure an accurate semi-analytical solution has been obtained, which can be used to compute displacements, stretches and stresses in each point of the cylinder. For the elliptical and rectangular sections, the results provided by the proposed solution method are shown by a series of graphs. Finally, the Poynting effect was investigated by varying the section shape of the cylinder.