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

This paper, in line with part I [53], is concerned with the finite element implementation of boundary potential energies and the study of their impact on the deformations of solids thereby the main thrust is the fully three-dimensional formulation and implementation incorporating anisotropic effects. Boundary effects can play a dominant role in the material behavior, the most prominent example being surface tension. However, the common modelling in continuum mechanics takes exclusively the bulk into account, nevertheless, neglecting possible contributions from the boundary. 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 normal. For the formulation of the finite element method, the concept of convected curvilinear coordinates attached to the boundary is employed and the corresponding 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. In contrast to the previous literature on this topic, the current manuscript covers jointly the following issues related to boundary energies: (1) the formulation and implementation represents a fully three-dimensional framework at large deformations, (2) the formulation of the problem and the proposed material models are based on finite strains, however, it is shown that the linearization would lead to the small strain models proposed previously in the literature and (3) the current manuscript covers the issue of anisotropy effects on the boundary energies which to the best of our knowledge has not been exemplified earlier.