A finite element framework for continua with boundary energies. Part I: The two-dimensional case

This contribution deals with the implications of boundary potential energies on the two-dimensional deformations of solids in the framework of the finite element method. Common modelling in continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the boundary. However, boundary effects sometimes play a dominant role in the material behavior, the most prominent example being surface tension. Within this contribution the boundary potentials are allowed, in general, to depend not only on the boundary deformation but also on the boundary deformation gradient and the spatial boundary tangent. For the finite element implementation, a suitable curvilinear convected coordinate system attached to the boundary is defined and corresponding geometrical and kinematical derivations completely based on a tensorial representation are carried out. Afterwards, the discretization of the generalized weak formulation, including boundary potentials, is performed and eventually numerical examples are presented to demonstrate the boundary effects due to different proposed material models