# On the Effect of a Rate-Dependent Work of Adhesion in the Detachment of a Dimpled Surface
## 1 Introduction  
Tribology is a very active field of research of utmost importance in several engineering applications, ranging from automotive [1] to aerospace [2] and bio-engineering [3]. In automotive, for example, the role of adhesion and viscoelasticity is crucial in determining tires' adherence and performance [4]. Adhesion due to van der Waals interactions is commonly exploited in nature, in which very efficient strategies have been developed to adhere to almost any kind of surface [5; 6]. Lizardw, geckos and insects in general in most cases outperform the most advanced human designed adhesive technologies, and the topic still has far to go. Geckos for example have developed a multiscale hierarchical structure, so that the \"macroscopic\" foot splits in several lamellae, which branch in the setae and finally in fine spatulae of the size of nanometers [7; 8]. Inspired by nature, several adhesive strategies have been pursued, among the others, that of fabricating patterned surfaces [9] present in mushroom pillars [10] or depressions (\"dimples\") [9] (see Figure 1a) and have been shown to be able to reach far higher adhesive stress with respect to the smooth case. Nevertheless, simple criteria to discern sticky from unsticky surfaces are still unavailable [11; 12; 13], except maybe the criterion introduced by [PERSON] at 3M, who suggested that a strong adhesive tape should have a Young's modulus lower than 0.3 MPa to be able to conform well to the harder counterpart [14], which, only recently, has received some support from a theoretical perspective [11; 12]. Over the years, several authors have conceived of contact mechanics models to ascertain how the enhancement is originated and possibly indicate routes for further development or better performance [15; 16; 17]. A very elegant model for the detachment of ahalfspace with a dimple from a flat substrate was proposed by [PERSON] et al. [17], who developed a contact mechanics model in the limit of short range adhesion (the so-called \"JKR limit\" from [PERSON], [PERSON], and [PERSON] seminal paper [18]) and showed that the dimple surface behaved as a pressure-sensitive adhesive: detaching the dimple from its equilibrium position would lead to \"weak adhesion\" in partial contact, but upon application of a compressive pressure, a full-contact state would be achieved that would require a (theoretically) infinite tensile traction to be detached (\"strong adhesion\"). In this respect, the finding recalls the seminal work of [PERSON] [19], who considered the contact of a halfplane with sinusoidal waviness. [PERSON] had to postulate the presence of a flaw at the interface (air trapping, contaminants, fine scale roughness...), which would guarantee the possibility to separate the two surfaces. [PERSON] et al. [15] have applied the [PERSON] et al. [17] model to study the adhesion capabilities of both cockroach pads (_Nauphoeta cinerea_) and dock beetles pads (_Gastrophivas viridula_), showing that they are strongly influenced by the geometric features of the interfacial pattern, and [PERSON] et al. [20] adopted the [PERSON] et al. [17] model to fit experimental results of adhesion of biomimetic polydimethylsiloxane (PDMS) surfaces patterned with pillars with mushroom-shaped tips. Recently, [PERSON] and [PERSON] [21] generalized the \"dimple\" model of [PERSON] et al. [17] by using a cohesive Maugis-Dugdale model, which introduces two new parameters, i.e., the theoretical strength of the material and the range of interaction of the adhesive tractions, and allowed investigation of the detachment performance of the dimple ranging from soft to hard materials as a function of a dimensionless parameter that is similar to that introduced by [PERSON] for the sphere [21; 22]. It was shown that the dimple adhesive performance, particularly the \"strong adhesive state\", becomes rapidly degraded, moving towards the rigid limit; hence for the dimple to effectively work as a pressure-sensitive mechanism, soft materials should be employed.