Figure 28.

The top panels show friction loops for three different contacts. ΔΓ is strictly proportional to the area inside the loop, divided by $u_{\mathrm{S}}^2$. Δf is nearly proportional to the maximum force divided by the maximum displacement (blue dots). The left-hand side (A) shows the linear viscoelastic contact. Δf and ΔΓ are constant. For partial slip (B) the ellipse-shaped loop turns into a lens-shaped loop. The area inside the loop increases with amplitude as $u_{\mathrm{S}}^3$ and $\Delta \mathsf{\Gamma }$ increases with amplitude, in consequence. The sketch in (B) is based on a quantitative model. The right-hand side (C) sketches the transition to gross slip. The diagram is motivated by experimental results [141], not by a quantitative model. $\Delta \mathsf{\Gamma }$ decreases at large amplitude because the friction force in steady sliding weakly depends on velocity. Dividing by $u_{\mathrm{S}}^2$ causes $\Delta \mathsf{\Gamma }$ to decrease with $u_{\mathrm{S}}$.