Fig. 3

Amontons’ law and the real contact area. a Real contact area vs. normal force. The area of real contact is obtained by thresholding the fluorescence images (Supplementary Fig. 1). Symbols show experiments on three PS spheres that have similar roughness. Solid lines show values obtained from theory as well as linear fits to the penetration hardness model, with p_Y the penetration hardness, and the fully elastic simulation, with p_rough the constant contact pressure. The inset shows the same data, but on a logarithmic scale. Experimental contact is reproduced by the contact hardening model that considers long-range elastic asperity interactions and local plasticity at contact. Other models either underestimate the contact area or do not describe the deviation from linearity found in the experiment. b Static friction force of contacts like those in a, measured at different normal forces. Symbols show experiments on two PS spheres, the red solid line is the hardening simulation fitted onto the friction data by multiplication with the interfacial shear strength. The agreement shows that the static friction force is proportional to the contact area. The constant of proportionality, or interfacial shear strength, is 50 MPa, close to the bulk shear strength of PS. Inset: the friction force F between a PS sphere and a glass substrate as a function of applied strain d measured using a rheometer. Through rotation of the rheometer plate (Fig. 1), a constant strain rate of ~1 μm s⁻¹ is imposed on the contact. The friction force builds up until slip occurs. The static friction is then defined as the maximal friction force at the onset of slip, measured at different normal forces, N, shown in the inset. Friction and contact data recorded during the event indicate that there is no stick slip behavior at the imposed sliding velocity