# On the Effect of a Rate-Dependent Work of Adhesion in the Detachment of a Dimpled Surface
## 4 Conclusions  
In this work, the detachment of a surface with an axisymmetric dimple from an elastic substrate with rate-dependent work of adhesion has been studied in the limit of short-range adhesion. Previous elastic model with constant work of adhesion has shown that the dimpled surface has two adhered states, one \"strong\" in full contact, one \"weak\" in partial contact. The rate dependence of the work of adhesion has been accounted for by adopting a power law dependence (with exponent \(n\)) of the effective work of adhesion on the pealing velocity as it is commonly observed in experiments [27]. Notice that, in this analysis, we have neglected bulk effects (we used the relaxed modulus of the viscoelastic material \(E_{0}=E_{\omega=0}\), with \(\omega\) being the excitation frequency of the material); hence, the results presented should be representative for not too high peeling velocity. It has been shown that a rate-dependent work of adhesion does not affect the \"strong\" adhesive state, but it strongly influences the detachment pull-off stress in partial contact, resulting in a strong toughening of the interface. In this respect, we have shown that for a given unloading rate, after a transient, the peeling velocity increases with a power law behavior during the detachment process, and so does the effective work of adhesion. Furthermore, it has been shown that the pull-off stress (in partial contact) does not depend much on the particular initial condition chosen to unload the dimple. Instead, the pull-off stress depends on three dimensionless parameters; i.e., it increases by increasing the unloading rate, the exponent \"\(n\)\" or the dimensionless parameter \(\alpha_{d}\). Clearly, for a real viscoelastic material, the interfacial toughening will not increase to infinity but will be limited by the ratio \(E_{\omega=\infty}/E_{\omega=0}\), which is known to vary anyway by orders of magnitude with the excitation frequency [26]. In this respect, for a real material, the curves in Figure 4 will be bounded. In conclusion, the presented results suggest that viscoelasticity may play a strong role in determining the apparent interfacial toughness in patterned interfaces. In particular, when soft rough substrates are considered, it may be very difficult to recognize other sources of toughening, such as those due to the increase of contact area due to roughness [30] or due the local jump instabilities as in the Guduru problem [31], unless an extremely low unloading rate is adopted as indeed recently attempted by [PERSON] and coauthors [32].  
A.P. acknowledges the support by the Italian Ministry of Education, University and Research under the Programme Department of Excellence Legge 232/2016 (Grant No. CUP-D94I 18000260001). 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 L2\"-D.D. n. 407, 27/02/2018, bando AIM (Grant No. AIM1895471). AP acknowledges support for the Open Access fees by Hamburg University of Technology (TUHH) in the funding programme Open Access Publishing.  
Not applicable.  
The author declares no conflict of interest.