**1. Introduction**

Adhesion is a challenging topic in tribology [1–3] with relevance in several engineering applications that range from biomimetics [4], soft matters [5], soft robots [6], grippers [7], friction [8–12]. Although roughness is usually responsible for adhesion reduction [13–15], Briggs and Briscoe [16] showed already in 1977 that relatively small random roughness amplitude could enhance adhesion in pull-off experiments as well as relative rolling resistance by a factor up to 2.5. Later, Guduru [17] showed that in the contact between a rigid sphere and a soft halfspace with an axisymmetric single wavelength waviness, adhesion could be enhanced by a factor up to 20 with respect to the Johnson–Kendall–Roberts smooth case ([18], JKR in the following). The enhancement was first modeled theoretically by Guduru [17] and then proved experimentally by Guduru and Bull [19]. The basic assumptions of the Guduru [17] model are that (i) the contact area is simply connected (there are no circular grooves within the contact patch) and that (ii) the halfspace is constituted by a soft material (elastomer or rubber) hence adhesion can be simply modeled by JKR theory [18]. Loading and unloading a rigid sphere from the wavy surface leads to several jump instabilities and related

dissipation, which is responsible for the measured enhancement. Kesari et al. [20] showed that if the roughness wavelength is substantially shorter than the sphere radius, then an envelope solution can be obtained, which describes well the loading-unloading hysteretical behavior well known to experimentalists (see also Kesari and Lew [21]).

Waters and coauthors in [22] developed a Maugis–Dugdale cohesive model, still based on the assumption of simply connected contact area, to account for the transition between the rigid and JKR limit. They showed that toughening and strengthening of the interface was mostly restricted to the JKR regime, while, in the rigid limit, they found the Bradley [23] solution for the smooth rigid sphere. Ciavarella [24] further discussed the assumptions of the Guduru model and the conclusions of Waters and coauthors [22]. In particular he noticed that for hard solids (i.e., in the rigid limit) the axisymmetric roughness should reduce the macroscopic adhesion by orders of magnitude with respect to the smooth sphere limit. Ciavarella [24] supported his argument by considering the Rumpf–Rabinowich model ([25–27]), which geometry is analogous but not equal to that of Guduru and is used for adhesion of hard particles (the model neglects the elastic deformation). The Rumpf–Rabinowich model predicts that increasing the substrate roughness the macroscopic adhesion force first decreases and then increases again. Ciavarella [24] suggested that the Guduru and the Rumpf–Rabinowich models may be respectively close to an upper and a lower bound for macroscopic adhesion of rough bodies (see also Ciavarella [28]).

In this paper, we reconsider the geometry of Guduru [17] and obtain a closed form solution for the rigid limit, which clearly shows that increasing the waviness amplitude *A* reduces the macroscopic adhesion force by orders of magnitude. By using the axisymmetric Boundary Element Method (BEM) the contact problem is solved with Lennard–Jones interaction law, for varying waviness amplitude *A* and wavelength *λ* and for different Tabor parameters of the sphere *µ*, without the restrictive assumption of a compact contact area. Numerical results are well in agreement with the theory both in the rigid and in the JKR limit. The transition from one regime to the other is numerically studied using the BEM code. In the JKR regime adhesion enhancement 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. It is shown that at large Tabor parameters *µ* (> 3), increasing *A* first leads to adhesion enhancement as predicted by Guduru theory [17], but then strongly reduces the macroscopic adhesive force due to the appearance of internal cracks. We found that for *A*/*λ* & 10−<sup>1</sup> the JKR solution greatly overestimates the pull-off force and the hysteretical dissipation.
