A rod model with thin-walled flexible cross-section: Extension to 3D motions and application to 3D foldings of tape springs
Résumé
A planar rod model with flexible cross-section has been proposed and discussed recently in literature (Guinot et al., 2012; Picault et al., 2014). This 1D model is especially suitable for the modeling of tape springs, which develop localized folds through the flattening of the cross-section, and represents a good alternative to 2D shell models which are numerically hard to perform. However, due to its planar nature, it does not account for the coupling between bending and twisting that is noticed experimentally. In this context, we propose an extension of this rod model to 3D motions that includes torsional warping of the cross-section, based on postulated assumptions that are approximations. Starting from a complete non-linear elastic shell model, we introduce an original kinematics inspired from the elastica model and from Vlassov’s theory (Vlassov, 1962) in order to describe in-plane and out-of-plane changes of the cross-section shape with few parameters. In the specific case of shallow tape springs, approximate expressions of the strain and kinetic energies are derived by performing an analytical integration over the cross-section. The 1D tape spring rod model eventually involves only eight kinematic variables: seven to describe the translation and the rotation of the cross-section (parameterized by a unit quaternion) and one for the shape of the cross-section which is assumed to remain circular. Expressions of the potential and kinetic energies are then introduced in a suitable finite element software that can perform an automatic differentiation to solve the problem. Two static cases are treated. The first one, a tape spring clamped at one end and submitted at the other end to a follower twisting moment, shows that the model account for the bending-twisting couplings in large displacements and large rotations. The second one, a novel test named the transverse bending test, illustrates the ability of the model to account for complex scenarios of 3D foldings, involving bending and twisting as well as the creation of folds, their migration along the tape and their duplication. Quantitative comparisons to numerical results obtained with a shell finite element model are shown.