MODELLING AND NUMERICAL SIMULATION OF STRINGS BASED ON LIE GROUPS AND ALGEBRAS APPLICATIONS TO THE NONLINEAR DYNAMICS OF REISSNER BEAMS

A nonlinear string can be modelled as a prestressed beam using Reissner’s assumptions. Namely, that plane sections normal to the neutral axis remain plane but that their displacements and rotations can be arbitrarily large. The strains are assumed to be small enough to neglect material nonlinearity, which means that each section is a rigid body. The nonlinear analysis of such a system can lead to a complex formulation when second Piola-Kirchoff stress and Green-Lagrange strain tensors are used. Alternatively, Lie groups and algebras offer a efficient formulation by considering the space of mechanical system transformations (the SE3 group) instead of the generalised co-ordinates space (R3). The intrinsic nonlinearities due to the curvature of the group SE3 (geometric nonlinearities) are correctly handled and this leads to a compact and exact form of the nonlinear equilibrium equations from which further models can easily be derived. As evidence, a linearisation in the neighbourhood of the prestressed beam can be written taking into account tension, flexion, shear, rotation and coupling phenomena. The general nonlinear problem can also be solved using pure numerical methods or semi-analytical Volterra series