# On the Effect of a Rate-Dependent Work of Adhesion in the Detachment of a Dimpled Surface
## 3 Results  
### Detachment Curves  
Let us investigate first the effect that a rate-dependent work of adhesion has on the detachment curve of the dimple by solving Equation (11). Figure 3 shows the unloading curves for the rate-dependent model (Equation (11), black dot-dashed curves) starting from different initial conditions \(\widehat{c}_{0}\) on the branch AB (Figure 2) for \(\alpha_{d}=0.5\), \(n=0.5\) and \(\dot{\widehat{\sigma}}_{A}=1\). The red curve (solid for the stable, dashed for the unstable branches) shows the [PERSON] elastic reference solution [17]. Clearly, the rate-dependence of the work of adhesion has a toughening effect (dot-dashed black curves). Figure 4a shows the dimensionless peeling velocity \(\widehat{v}_{p}=v_{p}/V_{0}\) as it changes during the detachment process, while panel (b) shows the corresponding dimensionless work of adhesion \(\widehat{w}=w/w_{0}\), while the crack propagates. The peeling process starts with a vanishing velocity; i.e., instantaneously, the remote stress increases without change of the crack radius \(\widehat{c}\), and this can be easily checked if one substitutes a solution of the [PERSON] elastic model (Equation (7)) into the rate-dependent model (Equation (11)). By further unloading the dimple surface \(\left(\hat{\sigma}_{A}>0\right)\), the peeling velocity \(\widehat{v}_{p}\) increases with a power law behavior, and so does the interfacial toughness. Notice that the slope of the \(\widehat{\sigma}_{A}(\widehat{c})\) curve is inversely proportional to the peeling velocity; hence, the latter decays with a power law. We define the pull-off detachment stress as \(\widehat{\sigma}_{A,pull}=\lim_{\varepsilon\rightarrow+\infty}\widehat{ \sigma}_{A}\). Furthermore, Figure 3 shows the crack retraction curve, when the dimple is forced into the unstable regime (\(\widehat{c}\leq 1\), blue dot-dashed curve). Apart from a different transient behavior, similarly to the elastic solution, the rate-dependent solution will jump into a full contact strong adhesive state. Figure 3 shows that the effect of the initial preload on the detachment stress \(\widehat{\sigma}_{A,pull}\) is marginal, particularly in light of the dependencies on the parameters \(\left\{n,\hat{\sigma}_{A},\alpha_{d}\right\}\) that we will show in the following. From now on, we will consider as initial condition only the more \"likely\" situation of unloading the dimple from its equilibrium position \(\widehat{\sigma}_{A}=0\), which belongs to the stable branch ABfor \(\alpha_{d}\leq 1\) (otherwise for \(\alpha_{d}>1\), the dimple naturally jumps into the full-contact strong adhesive state). Let us look at the influence of the exponent \(n\) and of the unloading rate \(\dot{\hat{\sigma}}_{A}\) on the dimple detachment curves. Figure 5a shows that starting from the dimple equilibrium position \"C\" for \(\alpha_{d}=0.5\), \(n=0.5\) and \(\hat{\sigma}_{A}=10^{[-3,0,1,2]}\), where black dot-dashed curves indicate the rate-dependent solutions, while the red curves show the reference elastic solution. Figure 5a shows that the unloading rate has a strong influence on the detachment stress \(\widehat{\sigma}_{A,pull}\) and that the elastic solution is approached only for the very low unloading rate \(\dot{\widehat{\sigma}}_{A}=10^{-3}\). Notice that, in this particular case, for the reference elastic solution, we have \(\left.\widehat{\sigma}_{A}\right|_{\max}\approx 0.23\), while in the a rate-dependent case \(\widehat{\sigma}_{A,pull}\approx 1.7\) at \(\dot{\widehat{\sigma}}_{A}=10^{2}\)  
Figure 4: (**a**) Dimensionless peeling velocity and (**b**) corresponding apparent work of adhesion \(\widehat{w}=w/w_{0}\) for the unloading curves in Figure 3.  
Figure 3: Unloading curves of the dimpled surface starting from different initial conditions \(\widehat{c}_{0}=[1.05,1.2,1.5,2,2.5]\) for \(\alpha_{d}=0.5\), \(n=0.5\) and \(\hat{\sigma}_{A}=1\). Red lines for the Griffith elastic solution, dot-dashed black lines for the rate-dependent model unloading curves.