#### 5.1. Effect of the Tabor Parameter

In the previous subsections we have discussed two limits of the Guduru contact problem: the JKR and the rigid limit. Here the transition from one limit to the other is investigated numerically by using the BEM introduced in

Section 3.1.

Figure 8 shows the pull-off force as a function of the Tabor parameter

$\mu $ for

${\lambda}^{\u2020}=20$, and

${A}^{\u2020}=\left[0.1,1,10\right]$. Small waviness amplitude

${A}^{\u2020}=0.1$ (red circles) slightly perturbs the solution of the smooth sphere. Indeed at low Tabor parameter the pull-off force is equal to

${\left|\overline{W}\right|}_{pull-off}\approx 1.75$, while at higher

$\mu $ it gets slightly larger than

$1.5.$ In all the range between

$\mu =0.01$ and

$\mu =5$ the pull-off force remains in between the rigid and JKR values (2 and 1.5 respectively). Increasing the waviness amplitude by a factor 10 (

${A}^{\u2020}=1$, green squares) completely changes the picture.

Figure 8 shows that there exist three distinct regimes: (i) the rigid, (ii) the transition and (iii) the JKR regime. The pull-off force remains very small and equal to the rigid limit (dot-dashed line) up to

$\mu \approx 0.25$, then starts to increase up to about

${\left|\overline{W}\right|}_{pull-off}\simeq 3.2$ for

$\mu \approx 1$ and for

$\mu >1$ tends to the JKR limit (Equation (

5), dashed line). By further increasing the waviness amplitude leads to smaller pull-off forces not only in the rigid limit, but also at large Tabor parameters

$\mu $. We have indicated in

Figure 8 that at

$\mu \simeq 5$ the JKR prediction of the pull-off force is

${\left|\overline{W}\right|}_{pull-off}\simeq 20,$ while numerical results give

${\left|\overline{W}\right|}_{pull-off}\simeq 1.1$.

Figure 9 shows respectively the dimensionless gap

H (a) and the corresponding tractions

P (b) for

${\lambda}^{\u2020}=20$,

${A}^{\u2020}=1$ and

$\mu =\left[0.15,0.67,5\right]$ (respectively solid, dotted, dot-dashed line) and

${A}^{\u2020}=10,$$\mu =5$ (dashed line) at the pull-off point. Focusing on the three curves corresponding at

${A}^{\u2020}=1$ one recognizes that at low Tabor parameter

$\left(\mu =0.15\right)$ the maximum tensile force is reached when the sphere first touches the waviness crest, while for high Tabor parameter (

$\mu =5,$ pink dot-dashed line) the typical pressure spike appears at the boundary of the contact patch. In the intermediate regime

$\left(\mu =0.67\right)$ the maximum pull-off force is reached when the second crest first touches the sphere. Nevertheless, the material is too rigid to deform and the gap remains large at the first throat providing small adhesive tractions. It is useful to compare the solutions obtained for

$\left(\mu ,{A}^{\u2020}\right)=\left(5,1\right)$ with those for

$\left(\mu ,{A}^{\u2020}\right)=\left(5,10\right).$ In the latter case

Figure 8 showed that JKR theory highly overestimates the pull-off force obtained numerically. Indeed,

Figure 9 shows that the contact patch is clustered on the waviness peaks and axisymmetric grooves (internal cracks) appear, which destroy the well known enhancement mechanism of the Guduru geometry.

To better study the effect of the waviness amplitude

${A}^{\u2020}$,

Figure 10 shows the dimensionless pull-off force in absolute value as a function of the ratio

$A/\lambda $ for

${\lambda}^{\u2020}=\left[5,20,30,50\right]$,

${R}^{\u2020}=[50,100,200]$ and for a fixed

$\mu =3$ (see legend therein). For each value of

$\lambda $ Equation (

5) was used to determine the pull-off force predicted by the JKR model (dashed black lines), while numerical results obtained with BEM are reported with markers (see legend in

Figure 10). For amplitude to wavelength ratio below

$A/\lambda \lesssim {10}^{-1}$ the numerical simulations and the theoretical results are in very good agreement. For very small waviness amplitude the JKR result for the smooth sphere is obtained

$\left({\left|\overline{W}\right|}_{pull-off}=1.5\right)$, while increasing

$A/\lambda $ adhesion enhancement takes place up to

${\left|\overline{W}\right|}_{pull-off}\approx 10$ for

${\lambda}^{\u2020}=50$. It appears that longer wavelengths foster adhesion enhancement. For

$A/\lambda \gtrsim {10}^{-1}$, the pull-off force suddenly decreases and, for larger

${\left|\overline{W}\right|}_{pull-off}$, decays approximately with a power law, without showing a clear threshold for “stickiness” (complete elimination of adhesion), contrary to other recent theories on random roughness [

40,

41]. It is shown that the sphere radius markedly influences the pull-off decay, but, in the parametric region explored, it slightly affects the threshold

$A/\lambda \simeq {10}^{-1}$ at which the abrupt transition from adhesion enhancement to reduction takes place.

In

Figure 11 we have replotted the data in

Figure 10 as effective adhesion energy

${\overline{w}}_{c,eff}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{w}_{c,eff}/{w}_{c}$ versus the Johnson parameter

${\alpha}_{KLJ}$. Indeed, based on Kesari and Lew [

21] envelope solution, Ciavarella [

24] showed that in the JKR regime the effective adhesive energy at pull-off depends only on the Johnson parameter

${\alpha}_{KLJ}$, i.e.,

which is shown as a solid blue line in

Figure 11. On the contrary, a competitive mechanism has been proposed by Persson and Tosatti [

13], which tends to reduce the effective adhesive energy due to surface roughness in randomly rough surfaces. Persson and Tosatti [

13] criterion reads

where

${A}_{app}$ is the apparent contact area,

${A}_{true}$ is the real contact area, increased due to the substrate roughness, and

${U}_{el}$ is the elastic strain energy stored at full contact. The real contact area

${A}_{true}$ can be written as [

13]

where

${a}_{app}$ is the apparent contact radius. Dividing Equation (

27) by

${A}_{app}=\pi {a}_{app}^{2}$, it can be derived that for large enough

${a}_{app}/\lambda $In

Figure 10 we obtained the largest enhancement of the pull-off force (up to a factor 10) at about

$A/\lambda \simeq {10}^{-1}$, where Equation (

28) would give

${A}_{true}/{A}_{app}\simeq 1.03$ (notice that

${\overline{w}}_{c,eff}=\frac{2}{3}{\left|\overline{W}\right|}_{pull-off}$), hence, in the following, we will neglect this contribution.

For a single scale waviness

hence

which is reported as a dot-dashed red line in

Figure 11. The numerical results we obtained, plotted with markers in

Figure 11, show that at large

${\alpha}_{KLJ}$ the numerical results we obtained closely follow Equation (

24). For smaller

${\alpha}_{KLJ}$, instead,

${\overline{w}}_{c,eff}$ drops suddenly and decays by further reducing

${\alpha}_{KLJ}$ with a strong dependence on the waviness wavelength and sphere radius. Instead, the Persson–Tosatti energetic argument for adhesion reduction seems to give a lower bound to the effective work of adhesion.

#### 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

$\mu =4,$${A}^{\u2020}=0.4$,

${R}^{\u2020}=100$ and

${\lambda}^{\u2020}=10$. 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

${\left|\overline{W}\right|}_{pull-off}\simeq 2$ 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)).

Nevertheless, the amount of dissipation is strongly influenced by the ratio

$A/\lambda $ and the results obtained by the JKR model (Equation (

5)) may be strongly misleading. In

Figure 13 the curve dimensionless normal load

$\overline{W}$ versus

$-{\Delta}^{\u2020}/\mu $ obtained numerically (red solid line) is plotted for the same parameters of

Figure 12 but for

${A}^{\u2020}=3$. 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.

#### 5.3. Adhesion Map

To clarify the effect of

A,

$\lambda $ and

$\mu $ on the pull-off force

${\left|\overline{W}\right|}_{pull-off}$ we fixed the sphere radius

${R}^{\u2020}=100$ and change

$\mu =\left[{10}^{-1},5\right]$ and

$A/\lambda =[{10}^{-3},{10}^{0}]$.

Figure 14 and

Figure 15 shows the contour plot of the pull-off force respectively for

${\lambda}^{\u2020}=20$ and

${\lambda}^{\u2020}=5$. One immediately notices that larger adhesive forces are reached with longer wavelengths.

Figure 14 shows that adhesion enhancement happens in a limited parameter region. For very low ratio

$A/\lambda $ the contact problem reduces to that of the smooth sphere on a smooth halfspace, hence by changing the Tabor parameter from

$\mu =0.01$ to

$\mu =5$ one moves from the Bradley

${\left|\overline{W}\right|}_{pull-off}=2$ to the JKR solution

${\left|\overline{W}\right|}_{pull-off}=1.5$. Increasing

$A/\lambda $ for small Tabor parameter (

$\mu <{10}^{-0.6}\approx 0.25$) leads to a strong reduction of the pull-off force, as indeed we are in the range where the rigid solution of the Guduru problem holds (cfr.

Section 2,

Figure 7). Notice that keeping

$\lambda $ constant and increasing

$A/\lambda $ leads to both increasing of

${A}^{\u2020}$ and

$\alpha $ in

Figure 7 heading to very strong reduction of the macroscopic pull-off force. Instead, if

$A/\lambda $ is increased at large Tabor parameter (

$\mu \gtrsim 0.25$ for

$\lambda =20$), adhesion enhancement takes place and high pull-off forces can be reached (in

Figure 14 up to

${\left|\overline{W}\right|}_{pull-off}\simeq 4$ for

$\mu \simeq 5$). Contrary to JKR theory predictions, further increasing of the amplitude to wavelength ratio

$A/\lambda $ does not lead to stronger adhesive forces, but adhesion is destroyed by roughness.