A locking-free finite element formulation and reduced models for geometrically exact Kirchhoff rods

In this work, we suggest a locking-free geometrically exact finite element formulation incorporating the modes of axial tension, torsion and bending of thin Kirchhoff beams with arbitrary initial curvatures. The proposed formulation has been designed in order to represent general load cases and three-dimensional problem settings in the geometrically nonlinear regime of large deformations. From this comprehensive theory, we not only derive a general beam model but also several reduced formulations, which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads. The advantages of these reduced models arise for example in terms of simplified finite element formulations, less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models. A second core topic of this publication is the treatment of membrane locking, which is a locking phenomenon predominantly occurring in highly slender curved structures, thus, exactly in the prime field of application for Kirchhoff theories. In order to address the membrane locking effect, we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as Assumed Natural Strains (ANS) or reduced integration. The effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples.