Lumped parameter models for adhesive contact mechanics

Adhesion is very common when the nanoscale or microscale solids come into contact, thus playing an important role in a variety of situations of interest. Due to the special nature of these problems, computational methods are often superior to analytical and experimental techniques for problems in adhesive contact mechanics. Finite element method is probably the most commonly used tool for such computational studies, due to its versatility and robustness. A major difficulty with these studies is the extremely high computational resource requirement. In this work, we propose two lumped parameter algorithms, which can significantly alleviate this issue. These algorithms provide results which approach the correct solution from either side, thereby providing upper and lower bounds to the correct solution, and thus, an estimate of the error involved in the analysis. Further, these algorithms can easily be incorporated into commercial finite element software and hence can be used more widely. Applicability of these algorithms is illustrated by studying the adhesive interactions between an elastic sphere and a rigid half-space, including the jump-to-contact instability caused by adhesion at small separations.