# On the Effect of a Rate-Dependent Work of Adhesion in the Detachment of a Dimpled Surface
## 2 Griffith Equilibrium Solution for a Dimple  
### Elastic Problem  
Here, the model introduced by [PERSON] et al. [17] for the detachment of two elastic half-spaces is summarized. The geometry consists of two elastic bodies (see Figure 1b), with Young's modulus and Poisson's ratio \(\{E_{1},v_{1}\}\) and \(\{E_{2},v_{2}\}\), one of those flat while the other with an axysimmetric dimple, whose geometry is defined by (see Figure 1b)  
\[\left\{ \begin{array}{cc}\delta=\frac{2}{\pi}\delta_{0}e^{\left(\frac{r}{b} \right)},&\frac{r}{b}\leq 1\\ \delta=\frac{2r}{\pi b}\delta_{0}\Big{[}e^{\left(\frac{b}{r}\right)}-\left(1 -\frac{b^{2}}{r^{2}}\right)\kappa\left(\frac{b}{r}\right)\Big{]},&\frac{r}{b}> 1\end{array} \right. \tag{2}\]  
where \(r\) is the radial coordinate, \(\delta_{0}\) is the dimple amplitude, \(b\) is the dimple radius, \(\kappa(\theta)\) and \(\varepsilon(\theta)\) are, respectively, the complete elliptical integral of first and second kind with argument \(\theta\). [PERSON] [28] has shown that a uniform axisymmetric pressure of magnitude \(p\) applied over a region \(r<b\) on the surface of a half-space produces surface displacements as in Equation (2); hence, to make the two half-spaces conform to each other, one would need within the dimple \((r\leq b)\) a tensile traction \(T\) equal to [28]  
\[T=\frac{E^{*}\delta_{0}}{2b},\qquad r\leq b \tag{3}\]  
where \(\frac{1}{E^{*}}=\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}\) is the composite elastic modulus. Hence, if a remote stress \(\sigma_{A}\) (\(>\)0, when tensile) is applied, the stress \(\sigma_{zz}\) at the interface of the two half-spaces will be  
\[\left\{ \begin{array}{cc}\sigma_{zz}=\sigma_{A}+\frac{E^{*}\delta_{0}}{2b}, &\frac{r}{b}\leq 1\\ \sigma_{zz}=\sigma_{A},&\frac{r}{b}>1\end{array} \right. \tag{4}\]  
The combination of the remote stress applied and the inner (constant) stress within the dimple can be easily studied in the framework of the Linear Elastic Fracture Mechanics (LEFM) theory, as the dimple behaves as an axisymmetric crack under internal pressure \(p(r)\)[29]. Let us define, for a crack of radius \(c\), the auxiliary function ([29], 3.114a) \(g(x)=\int_{0}^{x}\frac{sp(s)ds}{\sqrt{x^{2}-s^{2}}}\)  
\[\left\{ \begin{array}{cc}g(x)=(\sigma_{A}+T)x,&x<b\\ g(x)=\sigma_{A}x+T\Big{(}x-\sqrt{x^{2}-b^{2}}\Big{)},&x>b\end{array} \right. \tag{5}\]  
Then, Equation (3.117) in [29] gives the stress intensity factor as \(K_{I}=\frac{2}{\sqrt{\pi c}}g(c)\), or equivalently, the energy release per unit area \(G=K_{I}^{2}/2E^{*}\)  
\[\left\{ \begin{array}{cc}G=\frac{2c}{\pi E^{*}}\Big{(}\sigma_{A}+\frac{E^{*} \delta_{0}}{2b}\Big{)}^{2},&\frac{c}{b}\leq 1\\ G=\frac{2c}{\pi E^{*}}\Bigg{[}\sigma_{A}+\frac{E^{*}\delta_{0}}{2b}\Bigg{(}1 -\sqrt{1-\left(\frac{b}{c}\right)^{2}}\Bigg{)}\Bigg{]}^{2}&\frac{c}{b}>1\end{array} \right. \tag{6}\]  
[PERSON] et al. [17] apply the classical Griffith energetic argument that the energy release per unit area \(G\) should be equal to the surface energy \(w_{0}\) of the contact pair (the \"toughness\"), leading to their LEFM model [17]. The equilibrium curves that link the remote tension to the crack radius are written in dimensionless form as  
\[\left\{ \begin{array}{ll}\widehat{\sigma}_{A}=-1+\alpha_{d}\sqrt{\frac{1}{ \varepsilon}},&\widehat{c}\leq 1\\ \widehat{\sigma}_{A}=-1+\alpha_{d}\sqrt{\frac{1}{\varepsilon}}+\sqrt{1-\left( \frac{1}{\widehat{c}}\right)^{2}},&\widehat{c}>1\end{array} \right. \tag{7}\]  
where we have introduced the following dimensionless parameters  
\[\alpha_{d}^{2}=\frac{2\pi w_{0}b}{E^{*}\delta_{0}^{2}},\qquad\widehat{\sigma}_ {A}=\frac{\sigma_{A}}{T},\qquad\widehat{c}=\frac{c}{b}, \tag{8}\]  
Notice that \(\alpha_{d}\) is analogous to the parameter that [PERSON] defined for the sinusoid [28] as it is proportional to the ratio between the adiabatic work of adhesion and the strain elastic energy to flatten the dimple.