Generalized Section-Section Interaction Potentials in the Geometrically Exact Beam Theory

Abstract

The present contribution proposes a universal framework to formulate generalized section-section interaction potentials (SSIP) in the geometrically exact beam theory and to derive the resulting section-section interaction force and moment laws in a variationally consistent manner. By exploiting the fundamental kinematic assumption of undeformable cross-sections, a description of section-section relative configurations via a minimal set of six (translational and rotational) relative coordinates is proposed. Based on work-pairing and the definition of section-section interaction forces and moments, a pair of objective (translational and rotational) deformation measures, either in spatial or in material form, is identified, which in turn allows for the definition of objective (i.e., frame-invariant) SSIP laws and an associated variational principle. As main application area for this theory, the modeling of inter-molecular interactions (e.g., due to electrostatic, van der Waals or repulsive steric forces) between slender fibers based on the geometrically exact beam theory is considered. Interestingly, it is shown that the force and moment stress resultants as well as the workconjugated deformation measures of the well-known Simo-Reissner beam theory can be identified as a special case of this general framework, where the interaction potential represents an internal stored-energy function and the interacting cross-sections are located at neighboring arc-length positions on one and the same beam. While originally derived for inter-molecular interactions, the proposed variational problem formulation is demonstrated to be of a very general nature, thus also allowing for the formulation of translational and rotational constraints between arbitrarily oriented cross-sections based on a penalty potential. In this context, a possible field of application is given by the modeling of complex fiber networks, potentially embedded in a matrix material, where the proposed approach can be employed for the formulation of coupling constraints between individual fibers or between fibers and the matrix phase.