Formulation and analysis of a three-dimensional finite element implementation for adhesive contact at the nanoscale

A three-dimensional finite element model for nanoscale contact problems with strong adhesion is presented. The contact description is based on the Lennard–Jones potential, which is suitable to describe van der Waals attraction between interacting bodies. The potential is incorporated into the framework of nonlinear continuum mechanics, and two different formulations, a body force (BF) and a surface force (SF) formulation, are derived. It is demonstrated that the model is highly accurate for contact surfaces where the minimum local curvature radius of the surface roughness is as low as 8 nm. The finite element implementation of the two formulations is provided and the overall contact algorithm is discussed. The numerical accuracy of the finite element discretization is analyzed in detail. It is shown that the SF formulation is more efficient than the BF formulation but loses accuracy as the strength of adhesion increases. The model has applications in computational biomechanics as is demonstrated by the computation of the adhesion of a gecko spatula.