A novel section-section potential for short-range interactions between plane beams

TL;DR

This work develops the improved section-section interaction potential (ISSIP) to accurately model short-range intermolecular forces between plane Bernoulli-Euler beams by introducing an offset between interacting cross sections via $q_1$ and a gap $q_2$. The ISSIP arises from a 6D-to-2D reduction of the interaction integral and yields a closed-form-like expression $\phi_{m,\mathrm{ss}}= c_{m,\mathrm{ss}} q_2^{-m+\frac{7}{2}} F(q_1,q_2,m)$ with a Gaussian hypergeometric factor, reducing computational cost while maintaining accuracy. Implemented within rotation-free BE isogeometric beams and validated against parallel cylinders and Lennard-Jones interactions, ISSIP outperforms the legacy LSSIP for nonzero offsets and aligns with SBIP results, while preserving symmetry when using an averaged local coordinate system. The study provides practical guidance on cutoffs and integration schemes, enabling robust simulations of peeling, adhesion, and pull-off phenomena in fibrous systems at micro- and nano-scales. Overall, ISSIP offers a principled, efficient pathway to simulate complex fiber interactions driven by short-range forces in engineering and bio-inspired applications.

Abstract

We derive a novel formulation for the interaction potential between deformable fibers due to short-range fields arising from intermolecular forces. The formulation improves the existing section-section interaction potential law for in-plane beams by considering an offset between interacting cross sections. The new law is asymptotically consistent, which is particularly beneficial for computationally demanding scenarios involving short-range interactions like van der Waals and steric forces. The formulation is implemented within a framework of rotation-free Bernoulli-Euler beams utilizing the isogeometric paradigm. The improved accuracy of the novel law is confirmed through thorough numerical studies. We apply the developed formulation to investigate the complex behavior observed during peeling and pull-off of elastic fibers interacting via the Lennard-Jones potential.

Figures (24)

• Figure 1: Model of intermolecular interaction. a) Interaction of two molecules via interaction potential. b) Coarse-graining of interaction between molecular assemblies.
• Figure 2: Degeneration of a planar 3D beam into an in-plane 1D beam model. Base vectors at the centroid and at an equidistant line are depicted.
• Figure 3: Interaction of two in-plane beams. a) Schematic global point-of-view. b) Enlarged area of two interacting parts of the beams, with colored interacting cross sections. c) Graphical projections of interacting cross sections, $x_s$ and $y_s$, into the plane $\gamma_x$ with the normal $\boldsymbol{t}_x$.
• Figure 4: Comparison of the section-section Lennard-Jones interaction potential. a) The results obtained with numerical integration and ISSIP for four values of offset $q_1$ vs. gap $q_2$. b) $L_2$--norm of the relative errors of LSSIP and ISSIP w.r.t. numerical integration vs. offset $q_1$.
• Figure 5: Replacement of section-section interaction forces by equivalent resultant forces at the centroids. a) Approximate resultant forces for short-range interactions ($q_2,q_1 \ll R_x,R_y$). b) Equivalent approximate resultant forces at the centroids.