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

• Published in: International journal of solids and structures 2023-08, Vol.276, p.112255, Article 112255
• Main Authors: Meier, Christoph, Grill, Maximilian J., Wall, Wolfgang A.
• Format: Article
• Language: English
• Subjects: Generalized section–section interaction potentials; Geometrically exact beam theory; Inter-molecular forces; Rotational constraints; Strain-energy function
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2023.112255

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). •Generalized interaction potentials are proposed for the geometrically exact beam theory.•A variational principle is stated based on objective generalized deformation measures.•A main application is given by the modeling of intermolecular forces between fibers.•Strain-energy functions represent an asymptotic limit of these interaction potentials.•As a special case, the interaction potentials can be used for constraint enforcement.