*Article* **Roughness-Induced Adhesive Hysteresis in Self-Affine Fractal Surfaces**

**Guido Violano \* and Luciano Afferrante**

Department of Mechanics, Mathematics and Management, Polytechnic University of Bari, Via E. Orabona, 4, 70125 Bari, Italy; luciano.afferrante@poliba.it

**\*** Correspondence: guido.violano@poliba.it

**Abstract:** It is known that in the presence of surface roughness, adhesion can lead to distinct paths of loading and unloading for the area–load and penetration–load relationships, thus causing hysteretic loss. Here, we investigate the effects that the surface roughness parameters have on such adhesive hysteresis loss. We focus on the frictionless normal contact between soft elastic bodies and, for this reason, we model adhesion according to Johnson, Kendall, and Roberts (JKR) theory. Hysteretic energy loss is found to increase linearly with the true area of contact, while the detachment force is negligibly influenced by the maximum applied load reached at the end of the loading phase. Moreover, for the micrometric roughness amplitude *h*rms considered in the present work, adhesion hysteresis is found to be affected by the shorter wavelengths of roughness. Specifically, hysteresis losses decrease with increasing fractal dimension and cut-off frequency of the roughness spectrum. However, we stress that a different behavior could occur in other ranges of roughness amplitude.

**Keywords:** adhesion hysteresis; rough surfaces; JKR theory

**Citation:** Violano, G.; Afferrante, L. Roughness-Induced Adhesive Hysteresis in Self-Affine Fractal Surfaces. *Lubricants* **2021**, *9*, 7. https://doi.org/10.3390/ lubricants9010007

Received: 1 December 2020 Accepted: 4 January 2021 Published: 7 January 2021

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#### **1. Introduction**

The hysteretic dissipation is given by the difference between the work needed to bring two bodies into contact and that required to detach them. Its origin may be related to various phenomena occurring at the contact interface. The main causes of hysteresis are viscoelasticity [1,2], plasticity [3], adhesive elastic instabilities at jump-in and jump-out of contact [4], and surface roughness [5]. In particular, all natural and artificial surfaces are rough at some scale. Therefore, hysteretic losses may affect several technological applications. For example, biomedical devices [6] and structural adhesives [7] must safely adhere to surfaces during their application, but they should be easy to remove for reuse. Moreover, a recent challenge in soft robotics is to create climbing robots with reversible adhesion skills [8].

In contact experiments on soft matter, velocity-dependent dissipations are usually measured during detachment as a consequence of bulk viscoelasticity [9]. In a recent work [5], Dalvi et al. carried out loading–unloading contact experiments between smooth silicone hemispheres and rough nanodiamond substrates. Their experiments were performed at very low velocities (60 nm/s) both for the approach and detachment. Such choice allows the avoidance of velocity-dependent dissipations. However, great adhesion hysteresis was still observed due to the roughness-induced increase in the true contact area. This effect is expected to occur in compliant materials with small root mean square (rms) roughness amplitude (*h*rms ≃ 1 nm) [1], when they are bring in complete contact. Moving from the assumption of full-contact conditions, Dalvi et al. applied Persson and Tosatti (PT) adhesion theory [10] for predicting the magnitude of adhesion hysteresis. They found that the hysteretic dissipation increases almost linearly with the true contact area *A*, and it is equal to the product between *A* and the intrinsic surface energy ∆*γ*, which depends on the interfacial adhesive properties of contacting bodies.

In [11], it is experimentally shown that *h*rms can both increase and decrease the effective adhesive surface energy with respect to the smooth case. Moreover, numerical simulations of continuum adhesive contacts [12] have shown that there is an optimal *h*rms that leads to a maximization of the hysteretic loss and pull-off force. Such value of *h*rms is found when the contact region turns from being simply connected to being multiply connected. Similarly, in [13] it is shown that the effective surface energy ∆*γ*eff reaches a maximum for a certain *h*rms that, for vanishing applied pressure, is quite close to the value above which the effective contact area *A* becomes smaller than the nominal one *A*0. Moreover, the enhancement in the adhesion for small *h*rms is much larger for *H* < 0.5, where *H* is the Hurst exponent, as the roughness-induced increase in the surface area is smaller when *H* > 0.5.

For RMS roughness amplitudes of the order of few microns, the true area of contact is expected to be predominantly multiply connected. In such case, partial contact conditions occur and surface roughness leads to a reduction in the true area of contact. This in turn destroys adhesion. However, Kesari et al. [11] found that adhesion hysteresis can also be measured for relative large *h*rms. Inspired by the experimental findings in [11], Deng and Kesari (DK) [14] developed an analytical model for estimating hysteresis losses in the adhesive elastic contacts under the assumption of large roughness. DK's model captures the increase of adhesion hysteresis with the penetration, which is usually called depthdependent hysteresis. Moreover, in this case, a linear increase of the hysteretic dissipation with the area of contact is observed.

Carbone et al. [15] developed a numerical code based on a Boundary Element Method (BEM) for predicting loading–unloading hysteresis loops in the adhesive elastic contact of fractal self-affine 1D rough profiles. Their simulations were conducted under partial contact conditions, with *A*/*A*<sup>0</sup> ranging from 0.25 up to 0.5. Due to adhesion hysteresis, two distinct paths were obtained for loading and unloading curves of the area vs. load relation. In particular, they found two sources of energy dissipation, one occurring at small scales and the second one at large scales.

In [16,17], similar multiasperity models have been developed to estimate adhesion hysteresis. They moved from the pioneering Greenwood and Williamson (GW) model, in which roughness is described by a distribution of identical spherical asperities. Adhesion is then implemented according to the classical theory of Johnson, Kendall, and Roberts (JKR) [18]. In a loading–unloading cycle, each asperity exhibits a hysteretic dissipation, which is due to jump-in and jump-off contact instabilities. The total adhesion hysteresis is returned by the contribution of each asperity. However, such models are based on a simplistic description of the surface roughness and do not take into account the elastic coupling between contact regions.

In this work, we propose an investigation of the adhesive elastic contact of rough surfaces, described by self-affine fractal geometries, with an advanced multiasperity model taking into account lateral interactions of asperities according to the authors of [19,20] and adhesion according to JKR theory. Moreover, the model takes also account of the jump-in and jump-off contact instabilities occurring on each asperity.
