*5.2. Hysteresis Cycle*

It is well known that in adhesive contact mechanics different loading paths can be followed in loading and unloading a contact pair, which leads also to hysteretical energy dissipation. Here we show how this gets affected by the waviness amplitude *A* by proposing two representative examples. In Figure 12 the loading curve obtained via BEM numerical simulation is plotted as a solid red line for *µ* = 4, *A* † = 0.4, *R* † = 100 and *λ* † = 10. On the same graph, the JKR loading curve for the smooth sphere (black dot-dashed curve) and for the Guduru geometry (blue dashed curve, Equation (5)) are plotted. Figure 12 shows that the numerical and the theoretical curves are very close each other and the maximum adhesive force reached is about *W pull*−*o f f* <sup>≃</sup> 2 giving a certain enhancement with respect to the smooth case. A possible loading path (in displacement control) is shown by the arrows. The jump-in and -out instability are labeled with numbers from "1" to "6" for the loading stage and

with letters from "a" to "f" during unloading. Looking at Figure 12 one sees the hysteretical dissipation (proportional to the area enclosed in the hysteretical loop in Figure 12), which could be well estimated by adopting the JKR model (Equation (5)).

**Figure 12.** The dimensionless normal load *<sup>W</sup>* is plotted versus −<sup>∆</sup> †/*µ*. The curve is obtained via BEM numerical simulation (solid red line) for *µ* = 4, *A* † = 0.4, *R* † = 100 and *λ* † = 10. The JKR curve for a smooth sphere (dot-dashed black line) and for the Guduru geometry (blue dashed line, Equation (5)) are also shown. Loading and unloading paths are indicated by arrows and the jump-in and -out contact points are respectively labeled by numbers from "1" to "6" and letters from "a" to "f".

Nevertheless, the amount of dissipation is strongly influenced by the ratio *A*/*λ* and the results obtained by the JKR model (Equation (5)) may be strongly misleading. In Figure 13 the curve dimensionless normal load *<sup>W</sup>* versus −<sup>∆</sup> †/*µ* obtained numerically (red solid line) is plotted for the same parameters of Figure 12 but for *A* † = 3. Together with the BEM numerical results the JKR curve for the smooth sphere (black dot-dashed line) and for the Guduru geometry (blue dashed line) are shown. One immediately recognizes that the JKR model (blue dashed line) is very far from the actual loading curve (solid red curve). While the sphere approaches the wavy halfspace the JKR model predicts very large fluctuations of the normal load and relative jumps from one branch to the other that would lead to very high energy dissipation. The BEM solution, instead, gives much smaller undulations of the loading curve and smaller jumps-in and -out contact.

**Figure 13.** The dimensionless normal load *<sup>W</sup>* is plotted versus −<sup>∆</sup> †/*µ*. The curve is obtained via BEM numerical simulation (solid red line) for *µ* = 4, *A* † = 3, *R* † = 100 and *λ* † = 10. The JKR curve for a smooth sphere (dot-dashed black line) and for the Guduru geometry (Equation (5)) are also shown. Loading and unloading paths are indicated by arrows and the jump-in and -out contact points are respectively labeled by numbers from "1" to "6" and letters from "a" to "f".
