Finite Element Formulations for Surface Tension Effects in Deformable Solids: With Focus on User Element (Uel) Development

Abstract

A novel finite element is developed to simulate the deformation of solids subjected to surface tension. The proposed formulations are implemented in the general-purpose finite element (FE) software ABAQUS, utilizing the user-defined element (UEL) option. The principle of virtual work is established for the Young-Laplace equation within a curvilinear coordinate system to account for surface tension effects. In addition to the conventional external virtual power, two supplementary external power contributions, attributed to surface and three-phase contact interfaces, are incorporated. Shape functions are introduced to calculate covariant and contravariant basis vectors, enabling the derivation of force vectors and tangent stiffness matrices within the FE formulations. The developed formulations are integrated into the UEL framework of commercial FE software ABAQUS. Validation is performed through three application problems with analytical solutions: (1) Laplace pressures of hyperelastic droplets, (2) eigenmode oscillations of droplets, and (3) wetting behavior of cuboid droplets. The simulation results demonstrate excellent agreement with the analytical solutions across all verification problems, accurately capturing various surface tension effects. Therefore, the FE framework utilizing the proposed formulations provides accurate and efficient analysis capabilities for deformation and stress in solids under surface tension, with potential extensions to more complex geometries and external loading scenarios.

Keywords: Surface tension, Finite element, User-defined element, Young-Laplace equation, Droplet oscillation, Droplet wetting