**6. Conclusions**

In this paper we have reconsidered the Guduru adhesive contact problem. The rigid solution has been derived, which has been shown to depend only on two parameters: the dimensionless waviness amplitude *A* † and the dimensionless parameter *α* = *AR*/*λ* 2 . It has been shown that increasing *A* † and *α* reduces the macroscopic pull-off force by orders of magnitude due to the effect of roughness. Secondly, by using a BEM numerical code with Lennard–Jones interaction law, we have investigated the effects of the waviness wavelength, amplitude and of the sphere Tabor parameter on the adhesion enhancement. It has been shown that adhesion enhancement is limited to a certain region of the plane *A*/*λ* versus *µ*. In particular, at low Tabor parameter increasing the ratio *A*/*λ* tends to destroy adhesion. For large Tabor parameters increasing the ratio *A*/*λ* first increases adhesion due to the Guduru enhancement mechanism, but later, for *A*/*λ* greater than about 10−<sup>1</sup> , the waviness amplitude gets too large, internal cracks appear and macroscopic adhesion reduces strongly. We have shown that in this region using the JKR model to estimate both the pull-off force and the dissipated energy by hysteresis leads to very large errors as the hypothesis of compact contact area does not hold.

The enhancement effect is well captured by the Johnson parameter as derived by Ciavarella–Kesari–Lew [21,24], and is much larger than the Persson–Tosatti enhancement [13] in terms of increase of real contact area due to roughness. The Persson–Tosatti energetic argument for adhesion reduction seems to give a lower bound to the effective work of adhesion.

The axisymmetric waviness in the Guduru contact problem is highly idealized with respect to more common randomly fractal roughness, hence it is difficult to give reasonable estimates of the parameters we have introduced in our model for a fractal randomly rough surface. The analysis made is intended to shed light into the problem of adhesion enhancement with a potential application to the development of nano- and micro-mechanical systems and of bioinspired adhesives. Experimental measurements have been reported by Santos et al. [42], which show how echinoderms' tube feet exploit adhesion enhancement to increase the interfacial toughness on rough substrates. Santos et al. [42] tried to explain the interfacial toughening accounting for an increased contact area obtained when the echinoderm feet conforms to the rough substrate. We have found that adhesion enhancement may be obtained also when the latter effect is negligible.

When rough surfaces are idealized by spherical caps, a very small radius of curvature is expected at the finest scale, which suggests asperity contact takes place at very low Tabor parameters, hence adhesion enhancement seems to be very unlikely. At present, the only viable route to adhesion enhancement seems to be the design of an ad-hoc macroscopic roughness profile.

**Author Contributions:** Conceptualization, A.P. and M.C.; Methodology, Validation, Investigation, Writing—original draft preparation, A.P.; writing—review and editing, A.P. and M.C.; Supervision: M.C.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** A.P. and M.C. acknowledge the support by the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP-D94I18000260001). A.P. is thankful to the DFG (German Research Foundation) for funding the project PA 3303/1-1. A.P. acknowledges support from "PON Ricerca e Innovazione 2014-2020-Azione I.2" - D.D. n. 407, 27/02/2018, bando AIM (Grant No. AIM1895471).

**Conflicts of Interest:** The authors declare no conflict of interest.

## **Appendix A. BEM Formulation with Constant Pressure Discrete Elements**

Equation (10) constitutes the nonlinear problem to be solved. A problem arises in evaluating the integral (11) as the kernel function *G* (*r*,*s*) is singular in *s* = *r*. The common approach is to discretize Equation (11) assuming that the pressure *σ* (*s*) has a simple form over a discrete element. To this end the simplest approach is to assume that the pressure is constant over each element. For a constant pressure *p* acting over the ring *c*<sup>1</sup> < *r* < *c*<sup>2</sup> the deflection at *r* of a single half-space is

$$
\mu\_z\left(r\right) = \frac{4\overline{p}}{\pi \mathcal{E}^\*} \left[ F\left(c\_2, r\right) - F\left(c\_1, r\right) \right] \tag{A1}
$$

where from Johnson [29]

$$F\left(c,r\right) = \begin{cases} \begin{array}{cc} c\mathbf{E}\left(\frac{r}{c}\right) \\ r\left[\mathbf{E}\left(\frac{c}{r}\right) - \left(1 - \left(\frac{c}{r}\right)^2\right)\mathbf{K}\left(\frac{c}{r}\right)\right] \end{array} & r \le c \\\ r\left[\mathbf{E}\left(\frac{c}{r}\right) - \left(1 - \left(\frac{c}{r}\right)^2\right)\mathbf{K}\left(\frac{c}{r}\right)\right] & r > c \end{array} \tag{A2}$$

being *K* (*k*) and *E* (*k*) respectively the complete elliptic integrals of first and second kind with modulus *k*.

Assume we have discretized the surface in *N* elements, so that we have *M* = *N* + 1 discretization points. The deflection at point *r<sup>i</sup>* due to a constant pressure *p<sup>j</sup>* ring in between the radii *r<sup>j</sup>* and *rj*+<sup>1</sup> is

$$\mu\_z\left(r\_{\bar{l}}\right) = \frac{4\overline{p}\_{\bar{j}}}{\pi \mathcal{E}^\*} \left[\mathcal{F}\left(r\_{\bar{j}+1}, r\_{\bar{l}}\right) - \mathcal{F}\left(r\_{\bar{j}}, r\_{\bar{l}}\right)\right] = \frac{1}{\mathcal{E}^\*} \mathcal{G}\_{l\bar{l}} \overline{p}\_{\bar{j}} \tag{A3}$$

$$\mathcal{G}\_{\rm ij} = \frac{4}{\pi} \left[ F\left(r\_{\rm j+1\prime}, r\_{\rm i}\right) - F\left(r\_{\rm j\prime}, r\_{\rm i}\right) \right] \tag{A4}$$

where the term *Gij* within square brackets depends only on the nodal coordinates, hence by varying *i*, *j* = 1, ..., *M* all the terms can be computed once for all.

#### **References**


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