Analytical and numerical investigations of linear and nonlinear torsional strains using position gradients.

The linear and nonlinear torsion-kinematic equations are developed in this study using the position-gradient vectors. It is shown that pure torsion of circular shafts leads to stretch and change of orientation of longitudinal fibers away from the shaft centerline. In addition to the stretch, there are two nonzero Green–Lagrange shear strains, demonstrating that torsion is not one of the independent continuum-mechanics shear modes. Consequently, in a continuum finite-element (FE) approach, representation of torsion by a rotation of a rigid cross section may not be sufficient to fully capture the torsion stretch and shear strains. In addition to the nonlinear-torsion formulation developed using the absolute nodal coordinate formulation (ANCF), the paper presents a new gradient interpretation of the linear-torsion formulation, develops its governing equations, and sheds light on its assumptions. The fundamental problems associated with using independent interpolation of finite rotations of rigid cross sections to describe the torsion are explained. It is demonstrated that the new general ANCF nonlinear-torsion formulation leads to the analytical solution when linearization assumptions are used. Application of moment and force-couple in flexible-body dynamics is discussed; and it is demonstrated that using an ad hoc approach with conventional infinitesimal-rotation finite elements leads to a linear torsion solution. Numerical examples are presented to compare linear and fully nonlinear solutions, and to show that one ANCF element is sufficient to obtain a solution in a good agreement with the analytical solution in the case of small torsion angles. In the case of large torsion angles, the analytical solution cannot always be considered as a reference solution since it is based on simplifying assumptions, which do not capture nonlinear and more general torsional behavior. [ABSTRACT FROM AUTHOR]