*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 *µ* for *λ* † = 20, and *A* † = [0.1, 1, 10]. Small waviness amplitude *A* † = 0.1 (red circles) slightly perturbs the solution of the smooth sphere. Indeed at low Tabor parameter the pull-off force is equal to *W pull*−*o f f* <sup>≈</sup> 1.75, while at higher *<sup>µ</sup>* it gets slightly larger than 1.5. In all the range between *µ* = 0.01 and *µ* = 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* † = 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 *µ* ≈ 0.25, then starts to increase up to about *W pull*−*o f f* <sup>≃</sup> 3.2 for *<sup>µ</sup>* <sup>≈</sup> 1 and for *<sup>µ</sup>* <sup>&</sup>gt; <sup>1</sup> 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 *µ*. We have indicated in Figure 8 that at *µ* ≃ 5 the JKR prediction of the pull-off force is *W pull*−*o f f* <sup>≃</sup> 20, while numerical results give *W pull*−*o f f* <sup>≃</sup> 1.1.

**Figure 8.** Pull-off force *W pull*−*o f f* (absolute value) as a function of the Tabor parameter *<sup>µ</sup>* for *λ* † = 20, *R* † = 100 and *A* † = [0.1, 1, 10] , respectively red circles, green squares and blue triangles. Dot-dashed lines mark the rigid limit (22) whil dashed lines the Johnson–Kendall–Roberts (JKR) limit (5). For *A* † = 10 at *µ* = 5 the JKR limit would give *W pull*−*o f f* <sup>≃</sup> 20.

Figure 9 shows respectively the dimensionless gap *H* (a) and the corresponding tractions *P* (b) for *λ* † = 20, *A* † = 1 and *µ* = [0.15, 0.67, 5] (respectively solid, dotted, dot-dashed line) and *A* † = 10, *µ* = 5 (dashed line) at the pull-off point. Focusing on the three curves corresponding at *A* † = 1 one recognizes that at low Tabor parameter (*µ* = 0.15) the maximum tensile force is reached when the sphere first touches the waviness crest, while for high Tabor parameter (*µ* = 5, pink dot-dashed line) the typical pressure spike appears at the boundary of the contact patch. In the intermediate regime (*µ* = 0.67) 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 *µ*, *A* † = (5, 1) with those for *µ*, *A* † = (5, 10). 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.

**Figure 9.** (**a**) Dimensionless gap *H* and (**b**) dimensionless tractions *P* versus the radial coordinate *r*/*ε* at the pull-off point for *λ* † = 20, *R* † = 100 and varying *µ* and *A* † (for both panels please refer to the legend placed in panel (**b**)).

To better study the effect of the waviness amplitude *A* † , Figure 10 shows the dimensionless pull-off force in absolute value as a function of the ratio *A*/*λ* for *λ* † = [5, 20, 30, 50], *R* † = [50, 100, 200] and for a fixed *µ* = 3 (see legend therein). For each value of *λ* 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*/*λ* . 10−<sup>1</sup> 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  *W pull*−*o f f* <sup>=</sup> 1.5 , while increasing *A*/*λ* adhesion enhancement takes place up to *W pull*−*o f f* <sup>≈</sup> 10 for *<sup>λ</sup>* † = 50. It appears that longer wavelengths foster adhesion enhancement. For *A*/*λ* & 10−<sup>1</sup> , the pull-off force suddenly decreases and, for larger *W pull*−*o f f* , 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 *<sup>A</sup>*/*<sup>λ</sup>* <sup>≃</sup> <sup>10</sup>−<sup>1</sup> at which the abrupt transition from adhesion enhancement to reduction takes place.

**Figure 10.** Dimensionless pull-off force in absolute value as a function of the ratio *A*/*λ* for the four cases *λ* † = [5, 20, 30, 50], *R* † = [50, 100, 200] and for a fixed *µ* = 3 (see legend). Dashed lines stand for the pull-off force predicted by the JKR model (Equation (5)), markers for BEM numerical results, while the dot-dashed line is a guide to the eye.

In Figure 11 we have replotted the data in Figure 10 as effective adhesion energy *wc*,*e f f* = *wc*,*e f f* /*w<sup>c</sup>* versus the Johnson parameter *αKL J*. 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 *αKL J*, i.e.,

$$
\overline{w}\_{c,eff} = \frac{2}{3} \left| \overline{W} \right|\_{pull-off} = \left( 1 + \frac{1}{\sqrt{\pi} a\_{\rm KLJ}} \right)^2 \tag{24}
$$

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

$$w\_{c,eff} = w\_c \frac{A\_{true}}{A\_{app}} - \frac{\mathcal{U}\_{el}}{A\_{app}} \tag{25}$$

where *Aapp* is the apparent contact area, *Atrue* is the real contact area, increased due to the substrate roughness, and *Uel* is the elastic strain energy stored at full contact. The real contact area *Atrue* can be written as [13]

$$A\_{true} = -2\pi \int\_{A\_{app}} dr r \left(1 + \frac{1}{2} \left|\nabla h\right|^2\right) \tag{26}$$

$$=-2\pi \int\_0^{a\_{app}} dr r \left(1 + \frac{1}{2} \left(\frac{2\pi A}{\lambda}\right)^2 \sin^2\left(\frac{2\pi r}{\lambda}\right)\right) \tag{27}$$

where *aapp* is the apparent contact radius. Dividing Equation (27) by *Aapp* = *πa* 2 *app*, it can be derived that for large enough *aapp*/*λ*

$$\frac{A\_{true}}{A\_{app}} \simeq 1 + \pi \left(\frac{A}{\lambda}\right)^2. \tag{28}$$

In Figure 10 we obtained the largest enhancement of the pull-off force (up to a factor 10) at about *<sup>A</sup>*/*<sup>λ</sup>* <sup>≃</sup> <sup>10</sup>−<sup>1</sup> , where Equation (28) would give *Atrue*/*Aapp* ≃ 1.03 (notice that *wc*,*e f f* = 2 3 *W pull*−*o f f*), hence, in the following, we will neglect this contribution.

For a single scale waviness

$$\frac{\mathcal{U}\_{el}}{A\_{app}} = \frac{E}{4\left(1 - \nu^2\right)} \int d^2 q q \mathcal{C}\left(q\right) = \frac{\pi E}{4\left(1 - \nu^2\right)} \frac{A^2}{\lambda} \tag{29}$$

hence

$$
\overline{w}\_{c,eff} = 1 - \frac{1}{2\pi} \frac{1}{a\_{\rm KLJ}^2} \tag{30}
$$

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 *αKL J* the numerical results we obtained closely follow Equation (24). For smaller *αKL J*, instead, *wc*,*e f f* drops suddenly and decays by further reducing *αKL J* 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.

**Figure 11.** The data showed in Figure 10 are reported here as effective adhesion energy *wc*,*e f f* = *wc*,*e f f* /∆*γ* versus the *αKL J* for the four cases *λ* † = [5, 20, 30, 50], *R* † = [50, 100, 200] and for a fixed *µ* = 3 (see legend). The dot-dashed line stands for the reduction criterion of Persson and Tosatti [13], the solid line for the enhancement criterion of Ciavarella [24] based on the Kesari and Lew [21] solution of the Guduru problem and the dashed line is a guide to the eye.
