Abstract
SummaryThis contribution proposes the first three‐dimensional (3D) beam‐beam interaction model for molecular interactions between curved slender fibers undergoing large deformations. While the general model is not restricted to a specific beam formulation, in the present work, it is combined with the geometrically exact beam theory and discretized via the finite element method. A direct evaluation of the total interaction potential for general 3D bodies requires the integration of contributions from molecule or charge distributions over the volumes of the interaction partners, leading to a six‐dimensional integral (two nested 3D integrals) that has to be solved numerically. Here, we propose a novel strategy to formulate reduced section‐section interaction laws for the resultant interaction potential between a pair of cross‐sections of two slender fibers such that only two one‐dimensional integrals along the fibers' length directions have to be solved numerically. This section‐section interaction potential (SSIP) approach yields a significant gain in efficiency, which is essential to enable the simulation of relevant time and length scales for many practical applications. In a first step, the generic structure of SSIP laws, which is suitable for the most general interaction scenario (eg, fibers with arbitrary cross‐section shape and inhomogeneous atomic/charge density within the cross‐section) is presented. Assuming circular, homogeneous cross‐sections, in a next step, specific analytical expressions for SSIP laws describing short‐range volume interactions (eg, van der Waals (vdW) or steric interactions) and long‐range surface interactions (eg, Coulomb interactions) are proposed. Besides ready‐to‐use expressions for the total interaction potential, also the resulting virtual work contributions, its finite element discretizations, as well as a suitable numerical regularization for the limit of zero separation are derived. The validity of the SSIP laws, as well as the accuracy and robustness of the general SSIP approach to beam‐beam interactions, is thoroughly verified by means of a set of numerical examples considering steric repulsion, electrostatic, or vdW adhesion.
Highlights
Biopolymer fibers such as actin, collagen, cellulose, and DNA and glass fibers or carbon nanotubes are ubiquitous examples for slender, deformable structures to be found on the scale of nanometers to micrometers
While the general model is not restricted to a specific beam formulation, in the present work, it is combined with the geometrically exact beam theory and discretized via the finite element method
First, this section reveals that full 6D numerical integration to compute the total interaction potential of slender continua is by orders of magnitude too expensive and can not reasonably be used as a numerical reference solution even in minimal examples of one pair of cylinders
Summary
Biopolymer fibers such as actin, collagen, cellulose, and DNA and glass fibers or carbon nanotubes are ubiquitous examples for slender, deformable structures to be found on the scale of nanometers to micrometers. On the other hand, studying the deformation of elastic, slender bodies has a long history in mechanics and today's geometrically exact finite element formulations for shear-deformable (Simo-Reissner) as well as shear-rigid (Kirchhoff-Love) beams have proven to be both highly accurate and efficient.[4,5,6] contact interaction between beams has been considered in a number of publications, see References 12-21 All these methods are motivated by the macroscopic perspective of nonpenetrating solid bodies rather than the microscopic view considering the first principles of intermolecular repulsive forces. While the general model is not restricted to a specific beam formulation, in the present work, it is combined with the geometrically exact beam theory and discretized via the finite element method This novel approach is based on reduced SSIP laws that describe the resulting interaction potential between a pair of cross-sections as a closed-form analytical expression.
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