A computational model for short-range van der Waals interactions between beams and shells

Abstract

We consider potential-based interactions between beams (or fibers) and shells (or membranes) using a coarse-grained approach with focus on van der Waals attraction and steric repulsion. The involved 6D integral over volumes of a beam and a shell is split into a 5D analytical pre-integration over the beam's cross section and a surrogate plate tangential to the closest point on the shell, and the remaining 1D numerical integration along the beam's axis. This general inverse-power interaction potential is added to the potential energies of the Bernoulli-Euler beam and the Kirchhoff-Love shell. The total potential energy is spatially discretized using isogeometric finite elements, and the nonlinear weak form of quasi-static equilibrium is solved using the continuation method. We provide error estimates and convergence analysis, together with two intriguing numerical examples. The developed approach provides excellent balance between accuracy and efficiency for small separations.

Figures (27)

• Figure 1: Beam-shell interaction. The interaction between a beam's disk-shaped cross section C and a shell is approximated as an interaction between disk C and an infinite plate, tangential to the shell at the closest point. The hat accent designates a quantity evaluated at the closest point.
• Figure 2: Schematic representation of the shell model in reference and current configurations.
• Figure 3: Schematic representation of the beam model in reference and current configurations.
• Figure 4: a) Disk-half-space interaction. b) Disk-plate interaction.
• Figure 5: Some types of interaction pairs considered for error estimates.

Theorems & Definitions (1)

• Remark