Generalized section–section interaction potentials in the geometrically exact beam theory: Modeling of intermolecular forces, asymptotic limit as strain-energy function, and formulation of rotational constraints

Abstract

The present contribution proposes a universal framework to formulate generalized section–section interaction potentials (SSIP) within the geometrically exact beam theory. By exploiting the fundamental kinematic assumption of undeformable cross-sections, an objective (i.e., frame-invariant) description of SSIPs via a minimal set of six (translational and rotational) relative coordinates, either in spatial or in material form, is proposed. Based on work-pairing, work-conjugated section–section interaction forces and moments, either in spatial or in material form, are identified that can be consistently derived from a variational principle. Interestingly, it is shown that hyperelastic stored-energy functions relating the deformation measures and stress-resultants of the well-known geometrically exact Simo–Reissner beam theory can also be identified as SSIPs when considering the asymptotic limit of small relative distances and rotations between the interacting cross-sections. Moreover, the proposed variational problem formulation is demonstrated to be of a very general nature, thus allowing for the formulation of translational and rotational constraints between arbitrarily oriented cross-sections based on either a penalty or a Lagrange multiplier potential. Possible applications include fiber-based structures and materials in technical and biological systems, where the proposed approach allows to model short- or long-ranged inter-molecular (e.g., electrostatic, van der Waals or repulsive steric) interactions between fibers in geometrically complex arrangements and to formulate translational and rotational coupling constraints between different fibers (e.g., cross-linked polymer chains) or between fibers and a matrix phase (e.g., fiber-reinforced composites).