Frictional Pinning of Amorphous Contacts with 2D and 3D Elastic Substrates

Disorder at the interface of an amorphous slider on a crystalline substrate leads to vanishing friction for large rigid contacts. The inclusion of elasticity introduces an energetic contribution that competes with the random interfacial potential and allows frictional pinning over a domain with characteristic radius a_c. The interfacial force scales linearly with a_c, while the elastic restoring force from deformation on the scale of a_c depends upon the dimension of the substrate. For a 2D substrate, the restoring force is independent of a_c, while it grows linearly with a_c in 3D. Comparison of the interfacial and elastic contributions shows that the pinning force is proportional to a_c in 2D. The 3D case is marginal because the forces scale with the same power of a_c. Extensive simulations are used to test the scaling of friction with disorder strength and system size. In 3D, we find an exponential decrease of the pinning force and increase in a_c with increasing substrate stiffness. In 2D, the expected power law scaling is observed. Substrates of finite thickness show a crossover in scaling with contact size that may be important in understanding solid lubricants with a plate structure.