Contact Mechanics of Rough Spheres: Crossover from Fractal to Hertzian Behavior

Figure 3

Normalized normal force 𝐹_𝑁 versus normal stiffness 𝑘 for a spherical indenter, superimposed with a random roughness (𝑅=0.5, 𝐻=0.7, 𝐿=1, ℎ_rms=9⋅10⁻³). The red-dotted line is the result of a 3D boundary element study. The black line is the theoretical prediction according to (16). Furthermore, the asymptotical behavior of the plain roughness (11) and the smooth Hertzian case (9) are shown. The slope of the asymptotes is 1/3 in the Hertzian case and (𝐻+1)⁻¹=0.5882 for the roughness. The crossover occurs at 𝐹_tr/(𝐿²𝐸^∗)=8×10⁻⁵; see (17).