Relationship between contact size and static friction: An approach for rigid crystalline surfaces

Relationship between contact size (A) and static friction (f) has been studied for rigid crystalline systems. We built a series of systems with two identical surfaces but different orientations and investigated the effects of the size and shape of the contact area on static friction. In these systems, there are numerous nontrivial commensurate contacts. Our results confirmed that the relationship between A and f was determined by both commensurability and shape of the contact. For commensurate contacts, $f\propto A$ independent of the shape. For incommensurate contacts, generally $f\propto A^0$ for regular shapes or $f\propto A^{1/4}$ for irregular shapes; however, in very few cases of regular shapes, $f\propto A^{1/2}$. Moreover, in above systems, commensurability of a contact can be easily changed by a perturbation of the misfit angle. Therefore, if the perturbation caused by the lateral force and the deformation of the surface are considered (as is the case in real systems), further research is necessary.