New analytical laws and applications of interaction potentials with a focus on van der Waals attraction

The paper aims to improve the efficiency of modeling interactions between slender deformable bodies that resemble the shape of fibers. Interaction potentials are modeled as inverse-power laws with respect to the point-pair distance, and the complete body-body potential is obtained by pairwise summation (integration). To speed-up integration, we consider the analytical pre-integration of potentials between specific geometries such as disks, cylinders, rectangles, and rectangular prisms. Several exact new interaction laws are obtained, such as disk-infinite half-space and (in-plane) rectangle-rectangle for an arbitrary exponent, and disk-disk and rectangle-rectangle for van der Waals attraction. To balance efficiency and accuracy, approximate laws are proposed for disk-disk, point-cylinder, and disk-cylinder interactions. Additionally, we have developed a novel formulation for the interaction between a spatial beam and an infinite half-space. The application of the pre-integrated interaction potentials within the finite element method is illustrated via two examples.