Projection vs. relaxation of adjacent bulk deformation for surface modeling: theoretical and numerical aspects.

Abstract Motivated by a strong coupling between bulk and surface physics, we present two approaches to derive continuum surface models from the three-dimensional, adjacent bulk deformation: projection and relaxation. In contrast to conventional ad hoc models, properties like the surface stress are consistently derived from a hyperelastic thermodynamic potential. While the projection approach captures classical in-plane stresses, the relaxation approach can further relax normal-normal and normal-shear coupling. A projection onto the surface is indeed always a relaxation if anisotropy is superficial, but not vice versa. The distinct behavior of projection and relaxation is theoretically discussed and highlighted by specific examples. Numerical implementation in a finite-element framework is subsequently elaborated. Its performance is illustrated by isotropic and anisotropic surface energies of a free cube and a beam under tension. Finally, both projection and relaxation constitute valuable alternatives to conventional surface models in terms of physics, geometry and computation.